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This paper deals with an initial-boundary value problem in two-dimensional smoothly bounded domains for the system n t + u · ∇ n = Δ n - ∇ · (n S (n) ∇ v) , x ∈ Ω , t > 0 , v t + u · ∇ v = Δ v - v + w , x ∈ Ω , t > 0 , w t + u · ∇ w = Δ w - w + n , x ∈ Ω , t > 0 , u t + κ (u · ∇) u + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ · u = 0 , x ∈ Ω , t > 0 , (∗) which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid, where κ ∈ R , the gravitational potential ϕ ∈ W 2 , ∞ (Ω) , and S (n) satisfies | S (n) | ≤ C S (1 + n) - α for all n ≥ 0 , C S > 0 and α > - 1. Under the boundary conditions (∇ n - n S (n) ∇ v) · ν = ∂ ν v = ∂ ν w = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , it is shown in this paper that suitable regularity assumptions on the initial data entail the following: (i) If α > - 1 and κ = 0 , then the simplified chemotaxis-Stokes system possesses a unique global classical solution which is bounded. (ii) If α ≥ 0 and κ ∈ R , then the full chemotaxis-Navier–Stokes system admits a unique global classical solution. [ABSTRACT FROM AUTHOR]

A solenoidal basis is constructed to compute velocities using a certain finite element method for the Stokes problem. The method is conforming, with piecewise linear velocity and piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet conditions are supported by constructing an interpolating operator into the solenoidal velocity space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the linear system for the velocity. A basis of the pressure space is also constructed that can be used to compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions have local support and lead to sparse linear systems. The basis construction is confirmed through rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite, which can be exploited to solve their corresponding linear systems. Significant efficiency gains over the usual saddle-point formulation are demonstrated computationally. [ABSTRACT FROM AUTHOR]

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics { (n 1) t + u ⋅ ∇ n 1 = Δ n 1 − χ ∇ ⋅ (n 1 ∇ w) + n 1 (λ 1 − μ 1 n 1 + a n 2) , x ∈ Ω , t > 0 , (n 2) t + u ⋅ ∇ n 2 = Δ n 2 + ξ ∇ ⋅ (n 2 ∇ z) + n 2 (λ 2 − μ 2 n 2 − b n 1) , x ∈ Ω , t > 0 , w t + u ⋅ ∇ w = Δ w − w + n 2 , x ∈ Ω , t > 0 , u ⋅ ∇ z = Δ z − z + n 1 , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + (n 1 + n 2) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under no-flux boundary conditions for n 1 , n 2 , w and z in three-dimensional bounded domains and no-slip boundary conditions for u , this is ∂ n 1 ∂ ν = ∂ n 2 ∂ ν = ∂ w ∂ ν = ∂ z ∂ ν = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , where χ > 0 , ξ > 0 , μ 1 ≥ 0 , μ 2 ≥ 0 , λ 1 ≥ 0 , λ 2 ≥ 0 , a ≥ 0 , b ≥ 0 and ϕ ∈ W 2 , ∞ (Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution. [ABSTRACT FROM AUTHOR]

Water has engaged in a series of activities during plant growth. Therefore, the moisture content is one of the most important indicators of plant growth. Water deficiency can lead to a decrease in the stomatal degree on the leaf surface, where the carbon dioxide can flow into the plant leaf. The resulting photosynthetic efficiency can greatly reduce the appearance, nutrition, and total biomass of forestry plants. However, it is still lacking in the rapid detection of moisture content in plants. In this study, a polarization imaging system was proposed using a split-focal plane polarization camera. Polarization images were then analyzed to establish the dependence between the linear polarization and moisture content, in order to reveal the interaction between polarized light and water in plants. The rapid and nondestructive detection was realized for the moisture content of wintergreen leaves. The specific research was as follows. 1) The holly leaves were first collected and dried in a gradient manner. The moisture content was obtained after weighing; 2) A sub-focal plane polarization system was developed to detect the polarization of the gradient-dried holly leaves; 3) A systematic investigation was carried out to clarify the influence of growth states on the moisture content of holly leaves under various months, different parts, as well as sunny or rainy weather conditions. 40 leaves were collected and then divided into 8 groups, with 5 leaves for the control in each group; 4) The holly leaves were detected for the polarization. Stokes parameters were obtained to convert into the linear polarization degrees with clear physical significance and imaging. It was found that the linear polarization degree increased gradually, as the moisture content decreased. There was a strong negative correlation between the leaf moisture content and the linear polarization degree. 5) Rapid detection of moisture content was achieved by the quantitative relationship between moisture content and linear polarization degree after fitting. The degree of linear polarization was only used to detect the moisture content of the target leaf. 6) Other holly leaves were collected and then substituted into the fitting model, in order to verify the prediction of the model. The results show that the moisture content of holly spear leaves was represented by the linear polarization degree in the three growth conditions (months, weather, and parts). Furthermore, the linear polarization degree increased gradually with the decrease in the gradient of moisture content. A prediction model was obtained using the large number of plant leaves. The correlation coefficient reached 0.85, and the MSE was only 0.45%. Verification through the leaf moisture detection system showed that the average relative error of the moisture detection system was 9.50%, and the RMSE was 4.53%. The true moisture content of the leaf was characterized by linear polarization degree. The findings can also provide optical applications in plant health detection. [ABSTRACT FROM AUTHOR]

We derive an ultraweak variational formulation of the quad-curl problem in two and three dimensions. We present a discontinuous Petrov–Galerkin (DPG) method for its approximation and prove its quasi-optimal convergence. We illustrate how this method can be applied to the Stokes problem in two dimensions, after an application of the curl operator to eliminate the pressure variable. In this way, DPG techniques known from Kirchhoff–Love plates can be used. We present an a priori error estimate that improves a previous approximation result for effective shear forces by using a less restrictive regularity assumption. Numerical experiments illustrate our findings. [ABSTRACT FROM AUTHOR]

The chemotaxis‐Stokes system nt+u·∇n=Δnϕ(c),ct+u·∇c=Δc−c+n,ut=Δu+∇P+n∇Φ,∇·u=0$$ \left\{\begin{array}{rcl}{n}_t+u\cdotp \nabla n& =& \Delta \left(n\phi (c)\right),\\ {}{c}_t+u\cdotp \nabla c& =& \Delta c-c+n,\\ {}{u}_t& =& \Delta u+\nabla P+n\nabla \Phi, \kern2em \nabla \cdotp u=0\end{array}\right. $$is considered along with homogeneous boundary conditions of no‐flux type for n$$ n $$ and c$$ c $$, and of Dirichlet type for u$$ u $$, in a smoothly bounded domain Ω⊂ℝ2$$ \Omega \subset {\mathrm{\mathbb{R}}}^2 $$. Under the assumption that Φ∈W2,∞(Ω)$$ \Phi \in {W}^{2,\infty}\left(\Omega \right) $$, that ϕ∈C0((0,∞))$$ \phi \in {C}^0\left(\left(0,\infty \right)\right) $$ is bounded on each of the intervals (ξ0,∞)$$ \left({\xi}_0,\infty \right) $$ with arbitrary ξ0>0$$ {\xi}_0>0 $$, and that with some kϕ>0$$ {k}_{\phi }>0 $$ and α>0$$ \alpha >0 $$, we have ϕ(ξ)≥kϕ·(ξ+1)−αfor allξ>0.$$ \phi \left(\xi \right)\ge {k}_{\phi}\cdotp {\left(\xi +1\right)}^{-\alpha}\kern2em \mathrm{for}\ \mathrm{all}\kern0.4em \xi >0. $$It is shown that for any suitably regular initial data, an associated initial‐boundary value problem admits a global very weak solution. [ABSTRACT FROM AUTHOR]

This paper expands the isoparametric framework to construct a stable, H 1 -conforming, and divergence-free method for the Stokes problem in two dimensions based on the Scott-Vogelius pair on Clough-Tocher splits. The pressure space is defined through composition, whereas the velocity space is constructed via a new divergence-preserving mapping that imposes full continuity across shared edges in the isoparametric mesh. Our construction is motivated by operators and spaces found in isoparametric C 1 finite element methods. We prove the method is stable, pressure-robust, and has optimal order convergence. Numerical experiments are provided which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The latest geoid model "HELLAS GEOID 2023" (HG2023) derived by the Hellenic Military Geographical Service is the most comprehensive model for the entire Hellenic area. Long-term gravity data, orthometric and geometric heights, seabed topography and a high-resolution digital terrain model were implemented in the calculations. Data evaluation and their accuracy estimation were of major importance to ascertain compatibility among data sources. In this study, data from neighboring countries were used as they were essential for the completeness of the model developed. The technique remove–compute–restore was adopted for the separation of the gravity signal in the heterogeneous data utilizing EIGEN 6C4 (full degree and order 2190), since the global geodynamic model was found to fit best in the Greek region. Mean gravity residual contribution to the local geoid model was calculated using Stokes' theorem in the frequency domain and implementing the Fourier transform using the Wang and Core modification. The final surface of the resulting gravimetric geoid was adapted to the existing height system of the State utilizing the Least Squares Collocation method by fitting points of known orthometric and geometric heights, distributed throughout Greece. The external accuracy of the estimated geoid model was estimated at the level of 6 cm. [ABSTRACT FROM AUTHOR]

This paper uses the principles of polarisation to calculate the Stokes component of the polarisation image. It designs the spatial weighted fusion algorithm, focusing on the research and evaluation of the accuracy and time complexity of the spatial weighted fusion algorithm in the polarisation image fusion processing. The spatial weighted image fusion algorithm is used to fuse the polarisation degree image, the polarisation angle image, and the light intensity image. Finally, the algorithm function is realised through software programming to verify the effect and accuracy of the algorithm. Experimental results show that the spatial weighted fusion algorithm can keep the pixel value of the polarisation source image unchanged, avoid image distortion, artefacts and ringing effects algorithm is stable and efficient, with low time complexity. [ABSTRACT FROM AUTHOR]

In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwards via a postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuška–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Nečas–Babuška theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart–Thomas elements of order k along with piecewise polynomials of degree ≤k. In addition, the latter yield approximate local conservation of momentum for all three equations involved. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the aforementioned balance-preserving properties and the applicability of the proposed family of methods. [ABSTRACT FROM AUTHOR]

Magnetic resonance imaging allows the measurement of the three-dimensional velocity field in blood flows. Therefore, several methods have been proposed to reconstruct the pressure field from such measurements using the incompressible Navier–Stokes equations, thereby avoiding the use of invasive technologies. However, those measurements are obtained at limited spatial resolution given by the voxel sizes in the image. In this paper, we propose a strategy for the convergence analysis of state-of-the-art pressure reconstruction methods. The methods analyzed are the so-called Pressure Poisson Estimator (PPE) and Stokes Estimator (STE). In both methods, the right-hand side corresponds to the terms that involving the field velocity in the Navier–Stokes equations, with a piecewise linear interpolation of the exact velocity. In the theoretical error analysis, we show that many terms of different order of convergence appear. These are certainly dominated by the lowest-order term, which in most cases stems from the interpolation of the velocity field. However, the numerical results in academic examples indicate that only the PPE may profit of increasing the polynomial order, and that the STE presents a higher accuracy than the PPE, but the interpolation order of the velocity field always prevails. Furthermore, we compare the pressure estimation methods on real MRI data, assessing the impact of different spatial resolutions and polynomial degrees on each method. Here, the results are consistent with the academic test cases in terms of sensitivity to polynomial order as well as the STE showing to be potentially more accurate when compared to reference pressure measurements. [ABSTRACT FROM AUTHOR]

The rising velocity of an air bubble in a non‐Newtonian shear‐thinning fluid at low Reynolds numbers is generally similar to the Newtonian case given by Stokes' law. However, when the shear‐thinning fluid is subject to mechanical oscillations, the rising velocity could significantly increase. Here, we present a series of experiments quantifying the rising velocity of single bubbles during shaking in very high‐viscosity (2,000–30,000 Pa·s) shear‐thinning silicone oils. Air bubbles (18–30 mm diameter) were injected in a tank mounted on a shaking table. The tank was horizontally oscillated, at accelerations between 0.4 and 2 g. We observed a small increase in the rising velocity of the shaking cases at our experimental conditions. The increase was larger when bubbles were large and accelerations were high. Larger accelerations experienced the largest observational errors and we emphasize the exploratory nature of our results. We also measured the change in bubble diameter during the oscillations and computed the shear rate at the bubble surface. Maximum shear rates were in the range of 0.04–0.08 s−1. At these shear rates, our analysis indicates that shear thinning behavior of our analog fluids is expected to be small and compete with elastic behavior. This transitional viscous/elastic regime helps explain the small and variable results of our experiments. Our results are relevant to the study of earthquake‐volcano interactions. Most crystal‐free silicate melts would exhibit a purely viscous, shear‐thinning behavior in a natural scenario. Seismically enhanced bubble rise could offer an explanation for the observed increased degassing and unrest following large earthquakes. Plain Language Summary: The speed at which air bubbles rise in a liquid is generally well‐known. However, in non‐Newtonian fluids (liquids with non‐constant viscosity), previous experiments have shown that this speed could be significantly increased by shaking the fluid container. In this study, we present experiments where we measure the rising speed of an air bubble in a viscous non‐Newtonian fluid while shaking it. We found that bubbles rose slightly faster during shaking, especially for strong shaking and/or large bubbles. However, the shaking of the table reduced the precision of our measurements, and our results need to be further validated. We were also able to measure how quickly the bubbles deformed during shaking. When the bubbles deform quickly, our fluids may behave like solids rather than liquids. This helps explain why our results are less striking than previous experiments. Our results can be applied to the study of volcanoes. Magma inside a volcano contains vapor bubbles. When an earthquake occurs close to a volcano, these bubbles may accelerate toward the surface and lead to more gas coming out of the volcano. Key Points: We measured the rising rate of bubbles in shear‐thinning fluids subject to mechanical shakingThe bubble rising velocity increases slightly during shaking versus non‐shaking experimentsNatural magmas are likely to exhibit shear‐thinning behavior and increased bubble rise during large earthquakes [ABSTRACT FROM AUTHOR]

We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O(h) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O(h−1), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates. [ABSTRACT FROM AUTHOR]

The Hadamard–Rybczynski equation describes the steady‐state buoyant rise velocity of an unconfined spherical bubble in a viscous liquid. This solution has been experimentally validated for the case where the liquid viscosity is held constant. Here, we extend this result for non‐isothermal conditions, by developing a solution for bubble position in which we account for the time‐dependent liquid viscosity, liquid and gas densities, and bubble radius. We validate this solution using experiments in which spherical bubbles are created in a molten silicate liquid by cutting gas cavities into glass sheets, which are stacked, then heated through the glass transition interval. The bubble‐bearing liquid, which has a strongly temperature‐dependent viscosity, is subjected to various heating and cooling programs such that the bubble rise velocity varies through the experiment. We find that our predictions match the final observed position of the bubble measured in blocks of cooled glass to within the experimental uncertainty, even after the application of a complex temperature–time pathway. We explore applications of this solution for industrial, artistic, and natural volcanological applied problems. [ABSTRACT FROM AUTHOR]

We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems. [ABSTRACT FROM AUTHOR]

In this paper, we aim to determine the effective permeability of biporous solids containing fractures in a computational homogenization framework. Precisely, at the local scale, we use the Brinkman law for the porous solids and the Stokes equations within fractures. The resolution of the Brinkman/Stokes coupled equations is performed with the fast Fourier transform iterative scheme on periodic unit cells. The role of the dimensions and orientation of the cracks is investigated. The simulation results are also compared with an analytical formulation based on Mori–Tanaka estimates. The findings indicate that, depending on the considered flow direction, the dimensions and orientation of cracks strongly affect the effective permeability of fractured porous media. Besides, we also determine the macroscopic permeability for a population of fractures with regular and random orientations, dimensions and shapes. The results are then given and discussed. [ABSTRACT FROM AUTHOR]

We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscous droplet migrating in a viscous fluid of a different viscosity. Further, we assume that the surface of the droplet is partially contaminated with a stagnant layer of surfactant (surface active agent which reduces the interfacial tension). We analyze the effects of the following phenomena associated with the thermocapillary migration of a droplet in a transient Stokes flow. The first is the influence of surfactant cap for an arbitrary cap angle which is partially coated on the droplet surface for both high and low surface Péclet number cases. The second is the influence of the energy changes associated with stretching and shrinkage of the interfacial area elements, when the droplet is in motion. It can be noted that for the vanishing cap angle, both high and low surface Péclet number limits reduce to the case of a pure thermocapillary migration of a droplet in a transient Stokes flow. For a given ambient flow, the migration of the droplet is controlled by the magnitude of the ambient velocity and the surface tension variations due to temperature and surfactant concentration. In particular, these surface tension variations balance the tangential stress balance. Considering axisymmetric transient Stokes flow, we obtain analytical solutions in two limiting cases, namely low and high surface Péclet number. This work considers linear variation of interfacial tension on both thermal and surfactant gradients. The main contribution is pertaining to the capillary drift and the corresponding surfactant transport on the droplet for an axisymmetric hydrodynamic as well as thermal and surfactant fields. We have analyzed the level curves corresponding to stream function and temperature fields, i.e., streamlines and isotherms for various parameters in order to develop a realistic picture of the migration pattern and the influence of thermal fields. We observe that the streamlines in the vicinity of rear end of the droplet show asymmetry due to the surfactant accumulation at that region. Increasing cap angle breaks the symmetry of the induced stream. It is seen that increasing values of nondimensional parameter that accounts for the stretching and shrinkage of the droplet surface immobilizes the surface and offers retardation to the migrating droplet. The variation of migration velocity with time suggests a control mechanism for the migration of the drop under external/surface gradients and hence may serve as a useful tool in applications like targeted drug delivery systems. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze, up to our knowledge for the first time, Banach spaces-based mixed variational formulations for nearly incompressible linear elasticity and Stokes models. Our interest in this subject is motivated by the respective need that arises from the solvability studies of nonlinear coupled problems in continuum mechanics that involve these equations. We consider pseudostress-based approaches in both cases and apply a suitable integration by parts formula for ad-hoc Sobolev spaces to derive the corresponding continuous schemes. We utilize known and new preliminary results, along with the Babuška-Brezzi theory in Banach spaces, to establish the well-posedness of the formulations for a particular range of the indexes of the Lebesgue spaces involved. Among the aforementioned new results from us, we highlight the construction of a particular operator mapping a tensor Lebesgue space into itself, and the generalization of a classical estimate in L 2 for deviatoric tensors, which plays a key role in the Hilbertian analysis of linear elasticity, to arbitrary Lebesgue spaces. No discrete analysis is performed in this work. [ABSTRACT FROM AUTHOR]

A chemotaxis‐Stokes system modeling coral fertilization under the effects of p‐Laplace diffusion is investigated in spatially three‐dimensional setting. By making appropriate use of certain conditional estimates for chemotactic signal gradient and the fluid velocity, we prove global existence and boundedness under a mild assumption that p>2, which extends previous results derived in [8]. [ABSTRACT FROM AUTHOR]

In this paper we analyse geometric multiscale models arising in the description of physiological flows such as blood flow in arteries or air flow in the bronchial tree. The geometrical complexity of the networks in which air/blood flows lead to a classical decomposition in two areas: a truncated 3D geometry corresponding to the largest contribution of the domain, and a 0D part connected to the 3D part, modelling air/blood flows in smaller airways/vessels. The fluid in the 3D part is described by the Stokes or the Navier–Stokes system which is coupled to 0D models or so-called Windkessel models. The resulting Navier–Stokes–Windkessel coupled system involves Neumann non-local boundary conditions that depends on the considered applications. We first show that the different types of Windkessel models share a similar formalism. Next we derive existence results and stability estimates for the continuous coupled Stokes–Windkessel or Navier–Stokes–Windkessel problem as well as stability estimates for the semi-discretized systems with either implicit or explicit treatment of the boundary conditions. In all the calculations, we pay a special attention to the dependency of the various constants and smallness conditions on the data with respect to the physical and numerical parameters. In particular we exhibit different kinds of behavior depending on the considered 0D model. Moreover even if no energy estimates can be derived in energy norms for the Navier–Stokes–Windkessel system, leading to possible and observed numerical instabilities for large applied pressures, we show that stability estimates for both the continuous and semi-discrete problems, can be obtained in appropriate norms for small enough data by introducing a new well chosen Stokes-like operator. These sufficient stability conditions on the data may give a hint on the order of magnitude of the data enabling stable computations without stabilization method for the problem. Numerical simulations illustrate some of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (| ∇ c | 2) ∇ c) − n m , c t + u ⋅ ∇ c = Δ c − c + m , m t + u ⋅ ∇ m = Δ m − m n , u t = Δ u + ∇ P + (n + m) ∇ Φ , ∇ ⋅ u = 0 which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where S is a given function fulfilling | S (σ) | ≤ K S (1 + σ) − θ 2 , σ ≥ 0 with some K S > 0 . Based on conditional estimates of the quantity c and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if θ > 0 , then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that ∥ n (⋅ , t) − n ∞ ∥ L ∞ (Ω) + ∥ c (⋅ , t) − m ∞ ∥ W 1 , ∞ (Ω) + ∥ m (⋅ , t) − m ∞ ∥ W 1 , ∞ (Ω) + ∥ u (⋅ , t) ∥ L ∞ (Ω) → 0 as t → ∞ with n ∞ : = 1 | Ω | { ∫ Ω n 0 − ∫ Ω m 0 } + and m ∞ : = 1 | Ω | { ∫ Ω m 0 − ∫ Ω n 0 } + . [ABSTRACT FROM AUTHOR]

The treatment of exhaust gas from large diesel engines is an important factor to affect environment. Because of its advantages in reaction efficiency and heat transfer efficiency, circulating fluidized bed (CFB) has shown great potential with regard to save energy and purify emissions from large diesel engines. This paper mainly investigated a three-dimensional modeling of the CFB, and a rolling function is applied to simulate its unsteady working conditions, which is called the rolling circulation fluidized bed (RCFB). Two-fluid model based on Euler-Euler approach is utilized to numerically simulate the gas-solid two phases flow within the RCFB. In order to give the comparable results between the present simulation and the previous experiment, the effect of Stokes number on particle phase distribution behaviors in the RCFB is investigated. By simulating the overall swing condition of the RCFB based on a dynamic grid, the effect of the swing condition on particle radial distribution uniformity in the riser is obtained. The reliability of quantitative modeling is verified based on the experimental results of the extant literature. Based on this simulation, the influence of the Stokes number on the radial distribution of particles as well as the relationship between the uniformity of the distribution and particle size is obtained. [ABSTRACT FROM AUTHOR]

This work presents a method to obtain RGB pulsed lasers (683/532/436 nm) with pulse energies above 100 mJ for green and red laser beams. The maximum pulse energy of the blue laser beam is 16.5 mJ, resulting in a conversion efficiency of 5.9 %, a pulse width of 5.6 ns, and a maximum peak power of 2.9 MW. The pump source is a 532 nm pulsed laser, and the Raman medium is low-pressure hydrogen. The authors achieved a 683 nm Stokes Raman laser through stimulated Raman scattering and a 436 nm anti-Stokes Raman laser (AS1) through a four-wave mixing process. The conversion efficiency of the blue laser beam was optimized by adjusting the H 2 pressure and focus configuration. Additionally, a theoretical model was developed to explore methods for improving the blue laser conversion efficiency. By implementing a Herriot multiple-pass cavity with a special coating, the conversion efficiency of the AS1 laser reached 17.5 %. In conclusion, this work provides a practical technique for efficiently obtaining RGB lasers and demonstrates the potential for generating lasers with shorter wavelengths than the pump lasers. [ABSTRACT FROM AUTHOR]

Most numerical approaches for coupling fluid flow with porous media flow rely either on Stokes equations in the fluid part of the domain and Darcy's law in the porous part, or on Brinkman's equation. In both cases, difficulties arise at the boundary between the two parts because the equations used in the porous part are not of Stokes type. In this paper, an alternative to Darcy's law is proposed for modeling flows in porous media. This alternative relies on equations of Stokes type where the permeability tensor is replaced by force and stress derivative tensors. Numerical procedures are presented to compute these tensors from simulations at pore scale. Simulations in domains containing both fluid and porous parts are finally conducted simply assuming continuity of velocity and pressure and hence without imposing any condition at the boundary between the two parts. Results show that the proposed method is accurate and hence a promising alternative to Darcy's law for problems involving both fluid and porous parts. • Alternative to Darcy's law relying on equations of Stokes type for modeling flows in porous media. • Permeability tensor replaced by force and stress derivatives tensors computed from simulations at pore scale. • Flows in domains containing both fluid and porous parts modeled without any boundary conditions between the two parts. [ABSTRACT FROM AUTHOR]

This manuscript is concerned with the Keller-Segel-Stokes system (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n F (| ∇ c | 2) ∇ c) , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P + n ∇ Φ , ∇ ⋅ u = 0 , under no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded three-dimensional domains, with given suitably regular functions F and Φ. Here in accordance with recent developments in the literature on refined modeling of chemotactic migration, the introduction of suitably decaying F is supposed to adequately account for saturation mechanisms that limit cross-diffusive fluxes near regions of large signal gradients. In the context of such nonlinearities which suitably generalize the prototype given by F (ξ) = K F (1 + ξ) − α 2 , ξ ≥ 0 , with K F > 0 , known results addressing a fluid-free parabolic-elliptic simplification of (⋆) have identified the value α c = 1 2 as critical with regard to the occurrence of blow-up in the sense that some exploding solutions can be found when α < 1 2 , whereas all suitably regular initial data give rise to global bounded solutions when α > 1 2. The intention of the present study consists in making sure that the latter feature of blow-up prevention by suitably strong flux limitation persists also in the more complex framework of the fully coupled chemotaxis-fluid system (⋆). To achieve this, as a secondary objective of possibly independent interest the manuscript separately establishes some conditional bounds for corresponding fluid fields and taxis gradients in a fairly general setting that particularly includes the subsystem of (⋆) concerned with the evolution of (c , u , P). These estimates relate respective regularity features to certain integrability properties of associated forcing terms, as in the context of (⋆) essentially represented by the quantity n. The application of this tool to the specific problem under consideration thereafter facilitates the derivation of a result on global existence of bounded classical solutions to (⋆) for widely arbitrary initial data actually within the entire range α > 1 2 , and by means of an argument which appears to be significantly condensed when compared to reasonings pursued in previous works concerned with related problems. [ABSTRACT FROM AUTHOR]

To advance our understanding of the movement of elastic microstructures in a viscous fluid, techniques that utilize available data to estimate model parameters are necessary. Here, we describe a Bayesian uncertainty quantification framework that is highly parallelizable, making parameter estimation tractable for complex fluid–structure interaction models. Using noisy in silico data for swimmers, we demonstrate the methodology’s robustness in estimating fluid and elastic swimmer parameters, along with their uncertainties. We identify correlations between model parameters and gain insight into emergent swimming trajectories of a single swimmer or a pair of swimmers. Our proposed framework can handle data with a spatiotemporal resolution representative of experiments, showing that this framework can be used to aid in the development of artificial micro-swimmers for biomedical applications, as well as gain a fundamental understanding of the range of parameters that allow for certain motility patterns. [ABSTRACT FROM AUTHOR]

The Keller-Segel-Stokes system { nt + u ⋅ ∇n = Δn − ∇ ⋅ (n∇c) + ρn − μnα, ct + u ⋅ ∇c = Δc − c + n, ut = Δu + ∇P − n∇Λ , ∇ ⋅ u = 0 (*), is considered in a bounded domain Ω ⊂ ℝ3 with smooth boundary, with parameters ρ ≥ 0, μ > 0 and α > 1, and with a given gravitational potential Λ ∈ W2,∞(Ω). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect. [ABSTRACT FROM AUTHOR]

The presence of macropores and fractures significantly affects the effective transport properties of porous solids such as concrete and rocks. The dimensions of the fractures are generally large behind that of the initial porosity, so that the problem contains two porosities. The influence of the macroporosity is studied in the homogenization framework by solving at the intermediate scale, that of the macropores, a coupled Darcy/Stokes problem with the Beavers–Joseph–Saffman (BJS) interface condition. We derive analytic expressions of the macroscopic permeability in the case of an isotropic permeable matrix containing spheroidal-shaped macropores. To this aim, we consider a representative volume element (RVE) on which uniform boundary conditions are considered for the velocity and pressure fields. The local problem is written as minimum principles; kinematic and static approaches are developed to derive rigorous bounds for the macroscopic permeability. Closed-form expressions of the longitudinal permeability (along the revolution axe of the spheroid) are determined by considering a simplified RVE constituted of two confocal spheroids. They depend on the volume fraction and the eccentricity of the spheroidal macropores, the scale factor between the two porosities and the slip coefficient of the BJS model. Illustrations show the influence of these parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with the effects of nonlinear diffusion on global solvability and stabilization in a variety of models which generalizes the following prototype { n t + u ⋅ ∇ n = ∇ ⋅ (n m − 1 ∇ n) − ∇ ⋅ (n ∇ c) − ρ n , c t + u ⋅ ∇ c = Δ c − c + ρ , ρ t + u ⋅ ∇ ρ = Δ ρ − ρ n , u t + ∇ P = Δ u + (n + ρ) ∇ ϕ , ∇ ⋅ u = 0 with a given function ϕ ∈ W 2 , ∞ (Ω) , where Ω ⊂ R 3 is a general bounded domain with smooth boundary. Based on an energy-type argument combined with maximal Sobolev regularity theory, we conclude that if m > 37 33 , an associated initial-boundary value problem admits at least one globally bounded weak solution which stabilizes toward the spatial homogeneous equilibrium (n ∞ , ρ ∞ , ρ ∞ , 0) in the sense that ‖ n (⋅ , t) − n ∞ ‖ L p (Ω) + ‖ c (⋅ , t) − ρ ∞ ‖ W 1 , ∞ (Ω) + ‖ ρ (⋅ , t) − ρ ∞ ‖ W 1 , ∞ (Ω) + ‖ u (⋅ , t) ‖ L ∞ (Ω) → 0 as t → ∞ for any p > 1 , where n ∞ : = 1 | Ω | { ∫ Ω n 0 − ∫ Ω ρ 0 } + and ρ ∞ : = 1 | Ω | { ∫ Ω ρ 0 − ∫ Ω n 0 } +. [ABSTRACT FROM AUTHOR]

Although we humans have known since the first smokey campfires of prehistory that our activities might alter our local surroundings, the nineteenth century saw the first indications that humankind might alter the global environment; what we currently know as anthropogenic climate change. We are now celebrating the bicentenaries of three figures with a hand in the birth of climate science. George Stokes, John Tyndall and John Ruskin were born in August 1819, August 1820 and February 1819, respectively. We look back from the perspective of two centuries following their births. We outline their contributions to climate science: understanding the equations of fluid motion and the recognition of the need to collect global weather data together with comprehending the role in regulating terrestrial temperature played by gases in the atmosphere. This knowledge was accompanied by fears of the Earth's regression to another ice age, together with others that industrialization was ruining humankind's health, morals and creativity. The former fears of global cooling were justified but seem strange now that the balance has tipped so far the other way towards global warming; the latter, on the other hand, today seem very prescient. This article is part of the theme issue 'Stokes at 200 (Part 1)'. [ABSTRACT FROM AUTHOR]

Lucasian Professor Sir George Gabriel Stokes was appointed joint-Secretary of the Royal Society in 1854, a post he held for the unprecedented period of 31 years, relinquishing the role when he succeeded T.H. Huxley as President in 1885. An eminent scientist of the Victorian era, Stokes explained fluorescence (he also coined the word) and his hydrodynamical formulae (the 'Navier–Stokes equations') remain ubiquitous today in the physics of any phenomenon involving fluid flows, from pipelines to glaciers to large-scale atmospheric perturbations. He also made seminal advances in optics and mathematics, and formulae that bear his name remain widely used today. The historiography however appears to understate Stokes's significant impact on science as unacknowledged collaborator on a wide range of scientific developments. His scientific peers regarded him as a mentor, advisor, designer of crucial experiments and, as editor of the Royal Society's scientific journals, arbiter of the standards of excellence in scientific communication to be attained before publication would be considered. Three brief case studies on Stokes's correspondence with Lord Kelvin, Sir William Crookes and the chemist Arthur Smithells exemplify how his impact was conveyed through the work of other scientists. This paper also begins consideration of why the character and worldview of Stokes led him to eschew personal reputation and profit for the sake of science and the Royal Society, and of how the development of the discipline of history of science has impacted on historiography relating to Stokes and others. This article is part of the theme issue 'Stokes at 200 (Part 1)'. [ABSTRACT FROM AUTHOR]

George Gabriel Stokes won the coveted title of Senior Wrangler in 1841, a year in which the examination papers for the Cambridge Mathematical Tripos were notoriously difficult. Coming top in the Mathematical Tripos was a notable achievement, but for Stokes it was a prize hard won after several years of preparation, and not only years spent at Cambridge. When Stokes arrived at Pembroke College, he had spent the previous two years at Bristol College, a school which prided itself on its success in preparing students for Oxford and Cambridge. This article follows Stokes' path to the senior wranglership, tracing his mathematical journey from his arrival in Bristol to the end of his final year of undergraduate study at Cambridge. This article is part of the theme issue 'Stokes at 200 (Part 1)'. [ABSTRACT FROM AUTHOR]

This paper extends the algorithm of Benner et al. (2016) [10] to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in W1,≼ and L≼ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions. [ABSTRACT FROM AUTHOR]

This paper presents a numerical solution of the coupled system of the time‐dependent Stokes and fully dynamic Biot equations. The numerical scheme is based on standard inf‐sup stable finite elements in space and the Backward Euler scheme in time. We establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. To handle realistic parameters which may cause nonphysical oscillations in the pore fluid pressure, a heuristic stabilization technique is considered. Numerical errors and convergence rates for smooth problems as well as tests on realistic material parameters are presented. [ABSTRACT FROM AUTHOR]

We apply the AUSGeoid data processing and computation methodologies to data provided for the International Height Reference System (IHRS) Colorado experiment as part of the International Association of Geodesy Joint Working Groups 0.1.2 and 2.2.2. This experiment is undertaken to test a range of different geoid computation methods from international research groups with a view to standardising these methods to form a set of conventions that can be established as an IHRS. The IHRS can realise an International Height Reference Frame to be used to study physical changes on and within the Earth. The Colorado experiment study site is much more mountainous (maximum height 4401 m) than the mostly flat Australian continent (maximum height 2228 m), and the available data over Colorado are different from Australian data (e.g. much more extensive airborne gravity coverage). Hence, we have tested and applied several modifications to the AUSGeoid approach, which had been tailored to the Australian situation. This includes different methods for the computation of terrain corrections, the gridding of terrestrial gravity data, the treatment of long-wavelength errors in the gravity anomaly grid and the combination of terrestrial and airborne data. A new method that has not previously been tested is the application of a spherical harmonic high-pass filter to residual anomalies. The results indicate that the AUSGeoid methods can successfully be used to compute a high accuracy geoid in challenging mountainous conditions. Modifications to the AUSGeoid approach lead to root-mean-square differences between geoid models up to ~ 0.028 m and agreement with GNSS-levelling data to ~ 0.044 m, but the benefits of these modifications cannot be rigorously assessed due to the limitation of the GNSS-levelling accuracy over the computation area. [ABSTRACT FROM AUTHOR]

The coupled chemotaxis fluid system n t = ∇ · ((n + 1) m - 1 ∇ n) - ∇ · (n (n + 1) m - α - 1 ∇ c) - u · ∇ n , (x , t) ∈ Ω × (0 , T) , c t = Δ c - c + n - u · ∇ c , (x , t) ∈ Ω × (0 , T) , u t = Δ u + ∇ P + n ∇ ϕ , ∇ · u = 0 , (x , t) ∈ Ω × (0 , T) , ∇ c · ν = ∇ n · ν = 0 , u = 0 , (x , t) ∈ ∂ Ω × (0 , T) , n (x , 0) = n 0 (x) , c (x , 0) = c 0 (x) , u (x , 0) = u 0 (x) x ∈ Ω , is considered in a bounded domain Ω ⊂ R 3 , with smooth boundary, where m , α ∈ R . It is known that in the fluid-free case ( u ≡ 0 ), the system admits a global bounded solution if α > 1 3 . The purpose of this paper is to overcome difficulties arising from the presence of fluid interaction and to show that the same conclusion holds if m > - 1 3 . In the case m ≤ - 1 3 and α > 1 3 , global existence of classical solution will be shown. [ABSTRACT FROM AUTHOR]

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes. [Display omitted] • General control-volume finite-element framework. • Construction of overlapping control volumes giving more flexibility. • Hybrid approach using a hierarchical splitting for the MINI element. • Detailed numerical investigations for various test cases. [ABSTRACT FROM AUTHOR]

Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes type are presented. In order to obtain numerically scalable algorithms, coarse spaces obtained from the generalized Dryja-Smith-Widlund (GDSW) approach are used. Numerical results of our parallel implementation are presented for various incompressible fluid flow problems. In particular, cases are considered where the problem cannot or should not be reduced using local static condensation, e.g., Stokes or Navier-Stokes problems with continuous pressure spaces. In the new monolithic preconditioners, the local overlapping problems and the coarse problem are saddle point problems with the same structure as the original problem. Our parallel implementation of these preconditioners is based on the fast and robust overlapping Schwarz (FROSch) library, which is part of the Trilinos package ShyLU. The implementation is essentially algebraic in the sense that, for the class of problems presented here, the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Further information about the geometry or the null space of the underlying problem can improve the performance compared to the default settings. Parallel scalability results for several thousand cores for Stokes and Navier-Stokes model problems are reported. Each of the local problems is solved using a direct solver in serial mode, whereas the coarse problem is solved using a direct solver in serial or message passing interface (MPI)-parallel mode or using an MPI-parallel iterative Krylov solver. [ABSTRACT FROM AUTHOR]

We prove that the a standard adaptive algorithm for the Taylor--Hood discretization of the stationary Stokes problem converges with optimal rate. This is done by developing an abstract framework for quite general problems, which allows us to prove general quasi-orthogonality proposed in [C. Carstensen et al., Comput. Math. Appl., 67 (2014), pp. 1195--1253]. This property is the main obstacle towards the optimality proof and therefore is the main focus of this work. The key ingredient is a new connection between the mentioned quasi-orthogonality and LU-factorization of infinite matrices. [ABSTRACT FROM AUTHOR]

Abstract We consider the following chemotaxis–Stokessystem with rotation n t = Δ n − ∇ ⋅ (n S (x , n , c) ⋅ ∇ c) − u ⋅ ∇ n , c t = Δ c − f (x , n , c) − u ⋅ ∇ c , u t = Δ u + ∇ P + n ∇ ϕ , ∇ ⋅ u = 0 in Ω × (0 , T) , subject to the non-flux boundary conditions for n and c , as well as the Dirichlet boundary condition for u , where the bounded smooth domain Ω ⊂ R 3 , the matrix-valued function S ∈ C 2 (Ω ̄ × [ 0 , ∞) 2 ; R 3 × 3) fulfills | S (x , n , c) | ≤ S 0 (c) (1 + n) θ for all (x , n , c) ∈ Ω ̄ × [ 0 , ∞) 2 with S 0 nondecreasing, and f ∈ C 1 (Ω ̄ × [ 0 , ∞) 2 ; R) satisfies 0 ≤ f (x , n , c) ≤ f 0 (c) (n + 1) with f 0 nondecreasing and f (x , n , 0) = 0. It was proved that for any θ > 0 , the initial–boundary value problem possesses a unique globally bounded classical solution. [ABSTRACT FROM AUTHOR]

Abstract Eulerian derivatives of shape functionals in shape optimization can be written in two formulations of boundary and volume integrals. The former is widely used in shape gradient descent algorithms. The latter holds more generally, although rarely being used numerically in literature. For shape functionals governed by the Stokes equation, we consider the mixed finite element approximations to the two types of shape gradients from corresponding Eulerian derivatives. The standard MINI and Taylor–Hood elements are employed to discretize the state equation, its adjoint and the resulting shape gradients. We present thorough convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the volume integral formula has superconvergence property. Numerical results are presented to verify the theory and show that the volume formulation is more accurate. Highlights • We present convergence analysis of approximate shape gradients in Stokes equation. • Both volume and boundary formulations of shape gradients are considered. • MINI and Taylor–Hood elements are used to discretize shape gradients. • The volume formulation convergences faster and offers better accuracy. [ABSTRACT FROM AUTHOR]

In this paper we propose a finite volume method to solve the coupled Stokes-Darcy problem using steady Stokes equations for the fluid region and Darcy equations for the porous region. At the contact interface between the fluid region and the porous media we imposed two conditions. The first one is the normal continuity of the velocity, while the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use schwarz domain decomposition method. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the properties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that these kinds of methods lead to arbitrarily ill-conditioned systems and poorly approximated fluxes on unfitted interfaces/boundaries. In order to solve these issues, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with nontrivial geometries. [ABSTRACT FROM AUTHOR]

In this work, we develop and analyse a novel Hybrid High-Order discretisation of the Brinkman problem. The method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures. Based on the discrete unknowns, we reconstruct inside each element a Stokes velocity one degree higher than face unknowns, and a Darcy velocity in the Raviart–Thomas–Nédélec space. These reconstructed velocities are respectively used to formulate the discrete versions of the Stokes and Darcy terms in the momentum equation, along with suitably designed penalty contributions. The proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local (Stokes- or Darcy-dominated) regimes, as identified by a dimensionless number which can be interpreted as a friction coefficient. The singular limit corresponding to the Darcy equation is also fully supported by the method. Numerical examples corroborate the theoretical results. This paper also contains two contributions whose interest goes beyond the specific method and application treated in this work: an investigation of the dependence of the constant in the second Korn inequality on star-shaped domains and its application to the study of the approximation properties of the strain projector in general Sobolev seminorms. [ABSTRACT FROM AUTHOR]

We construct several stable finite element pairs for the Stokes problem on barycentric refinements in arbitrary dimensions. A key feature of the spaces is that the divergence maps the discrete velocity space onto the discrete pressure space; thus, when applied to models of incompress-ible ows, the pairs yield divergence-free velocity approximations. The key result is a local inf-sup stability that holds for any dimension and for any polynomial degree. With this result, we construct global divergence-free and stable pairs in arbitrary dimension and for any polynomial degree. [ABSTRACT FROM AUTHOR]

This paper deals with the following chemotaxis-Stokes system n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n f ( c ) , x ∈ Ω , t > 0 , u t = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R 2 with smooth boundary, ϕ ∈ W 1 , ∞ ( Ω ) , f and S are given sufficiently smooth functions with values in [ 0 , ∞ ) and R 2 × 2 , respectively. Here S satisfies | S ( x , n , c ) | < S 0 ( c ) n α with α ≥ 0 and some nondecreasing nonnegative function S 0 . It is showed that when m > 1 + α and α ≥ 0 , the corresponding system possesses a global bounded weak solution for any sufficiently regular initial data ( n 0 , c 0 , u 0 ) satisfying n 0 ≥ 0 and c 0 ≥ 0 . [ABSTRACT FROM AUTHOR]

A Skeleton-stabilized IsoGeometric Analysis (SIGA) technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed method allows utilizing identical finite dimensional spaces (with arbitrary B-splines/NURBS order and regularity) for the approximation of the pressure and velocity components. The key idea is to stabilize the jumps of high-order derivatives of variables over the skeleton of the mesh. For B-splines/NURBS basis functions of degree k with C α -regularity ( 0 ≤ α < k ), only the derivative of order α + 1 has to be controlled. This stabilization technique thus can be viewed as a high-regularity generalization of the (Continuous) Interior-Penalty Finite Element Method. Numerical experiments are performed for the Stokes and Navier–Stokes equations in two and three dimensions. Oscillation-free solutions and optimal convergence rates are obtained. In terms of the sparsity pattern of the algebraic system, we demonstrate that the block matrix associated with the stabilization term has a considerably smaller bandwidth when using B-splines than when using Lagrange basis functions, even in the case of C 0 -continuity. This important property makes the proposed isogeometric framework practical from a computational effort point of view. [ABSTRACT FROM AUTHOR]

Three first-order finite volume element methods, namely, conforming, nonconforming, and discontinuous Galerkin schemes for Stokes equations, are analyzed and compared using the medius analysis. The latter analysis is based on a combination of arguments from a priori and a posteriori error analyses under no extra regularity assumptions on the weak solution. Best-approximation results for the energy norms hold in all cases and allow for comparison results up to generic equivalence constants and higher-order data oscillation of the applied volume forces with little modification for different pressure approximations. A priori and a posteriori analyses of discontinuous Galerkin finite volume element methods with SIPG, IIPG, NIPG are included for completeness. [ABSTRACT FROM AUTHOR]