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We present some rigorous results about the bending behaviour of paper. By adapting these results to the Möbius strip, we obtain some qualitative properties of developable Möbius strips which minimize the bending energy. We also provide some numerical simulations which illustrate and strengthen the analytic results. [ABSTRACT FROM AUTHOR]

We propose to use a hybridizable discontinuous Galerkin (HDG) method combined with the continuous Galerkin (CG) method to approximate Maxwell's equations. We make two contributions in this paper. First, even though there are many papers using HDG methods to approximate Maxwell's equations, to our knowledge they all assume that the coefficients are smooth (or constant). Here, we derive optimal convergence estimates for our HDG-CG approximation when the electromagnetic coefficients are piecewise W 1 , ∞ . This requires new techniques of analysis. Second, we use CG elements to approximate the Lagrange multiplier used to enforce the divergence condition and we obtain a discrete system in which we can decouple the discrete Lagrange multiplier. Because we are using a continuous Lagrange multiplier space, the number of degrees of freedom devoted to this are less than for other HDG methods. We present numerical experiments to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart ( m = 1 ) or Morley ( m = 2 ) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated L 2 error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh T of the domain Ω , we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in L 2 and also on the appropriate subspace consisting of k-forms with exterior derivatives in L 2 . [ABSTRACT FROM AUTHOR]

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order k - 1 . It employs a "singular distance" (measured by some geometric mesh quantity Θ z ≥ 0 for triangle vertices z ) and imposes a local side condition on the pressure space associated to vertices z with Θ z = 0 . The method is inf-sup stable for any fixed regular triangulation and k ≥ 4 . However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0 < Θ z ≪ 1 . In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the "singular distance". We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence. [ABSTRACT FROM AUTHOR]

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L 2 -norm, the Galerkin projection onto this generalized finite element space even yields ε -independent error bounds, ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. More precisely, we consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in the L 2 , H 1 , H 1 + α and H 2 norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a linearized decoupled mass and energy conservation Crank-Nicolson (CN) fully-discrete scheme is proposed for the coupled nonlinear Schrödinger (CNLS) system with the conforming bilinear Galerkin finite element method (FEM), and the unconditional supercloseness and superconvergence error estimates in H 1 -norm are deduced rigorously. Firstly, with the aid of the popular time-space splitting technique, that is, by introducing a suitable time discrete system, the error is divided into two parts, the time error and spatial error, the boundedness of numerical solution in L ∞ -norm is derived strictly without any constraint between the mesh size h and the time step τ . Then, thanks to the high accuracy result between the interpolation and Ritz projection, the unconditional superclose error estimate is obtained, and the corresponding unconditional superconvergence result is acquired through the interpolation post-processing technique. At last, some numerical results are supplied to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the orthogonal Gauss collocation method (OGCM) with an arbitrary polynomial degree for the numerical solution of a two-dimensional (2D) fourth-order subdiffusion model. This numerical method involves solving a coupled system of partial differential equations by using OGCM in space together with the L1 scheme in time on a graded mesh. The approximations w h n and v h n of w (· , t n) and Δ w (· , t n) are constructed. The stability of w h n and v h n are proved, and the a priori bounds of ‖ w h n ‖ and ‖ v h n ‖ are established, remaining α -robust as α → 1 - . Then, the error ‖ w (· , t n) - w h n ‖ and ‖ Δ w (· , t n) - v h n ‖ are estimated with α -robust at each time level. In addition, superconvergence results of the first-order and second-order derivative approximations are proved. These new error bounds are desirable and natural, as that they are optimal in both temporal and spatial mesh parameters for each fixed α . Finally some numerical results are provided to support our theoretical findings. [ABSTRACT FROM AUTHOR]

The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities. [ABSTRACT FROM AUTHOR]

Many neurodegenerative diseases are connected to the spreading of misfolded prionic proteins. In this paper, we analyse the process of misfolding and spreading of both α -synuclein and Amyloid- β , related to Parkinson’s and Alzheimer’s diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on ϑ - method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guarantees that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of α -synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid- β in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson’s and Alzheimer’s diseases. [ABSTRACT FROM AUTHOR]

This paper develops efficient preconditioned iterative solvers for incompressible flow problems discretised by an enriched Taylor–Hood mixed approximation, in which the usual pressure space is augmented by a piecewise constant pressure to ensure local mass conservation. This enrichment process causes over-specification of the pressure when the pressure space is defined by the union of standard Taylor–Hood basis functions and piecewise constant pressure basis functions, which complicates the design and implementation of efficient solvers for the resulting linear systems. We first describe the impact of this choice of pressure space specification on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with preconditioners based on the singular pressure mass matrix, and for Oseen systems arising from linearised Navier–Stokes equations, by using a two-stage pressure convection–diffusion strategy. The codes used to generate the numerical results are available online. [ABSTRACT FROM AUTHOR]

In this paper, the first family of conforming finite element divdiv complexes on cuboid meshes is constructed. The complex exhibits exactness on a contractible domain in the sense that the kernel space of each successive discrete map is the range of the previous one. This allows for algebraic structure-preserving finite element discretization of both the biharmonic equation and the linearized Einstein–Bianchi system. The convergence of optimal order is established and validated through numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we locally construct a conservative flux for finite element solutions of elliptic interface problems with discontinuous coefficients. Since the Discontinuous Galerkin method has built-in conservative flux, we consider in this paper the conforming finite element method and a special type of nonconforming method with arbitrary orders. We also perform our analysis based on Nitsche’s method, which imposes the Dirichlet boundary condition weakly. The construction method is derived based on a mixed problem with one solution coinciding with the finite element solution and with the other solution being naturally used to obtain a conservative flux. We then apply the recovered flux to the a posteriori error estimation and prove the robust reliability and efficiency for conforming elements, under the assumption that the diffusion coefficient is quasi-monotone. Numerical experiments are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the decoupled, positivity-preserving and unconditionally energy stable fully discrete finite element schemes for the electrohydrodynamic flow with variable density. After deriving some new features of the nonlinear coupled terms, by introducing scalar auxiliary variable methods to ensure the positivity and the boundedness of the approximation fluid density, we construct linear and decoupled first- and second-order fully discrete least square finite element methods for the model. Compared with the classical ones, not only the positivity-preserving technique in the proposed methods has the form invariance and is independent of the discrete methodology, but also much better computational cost and accuracy can be achieved. Moreover, by proposing modified zero-energy-contribution methods to balance the errors generated in the decoupled processes for the nonlinear coupled terms, we prove that two fully discrete schemes are unconditionally energy stable. The shown numerical examples confirm the superiority in the computational time, the positivity-preserving, the stability and the computational accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the optimal convergence order of a weak Galerkin finite element approximation to a second-order parabolic equation whose solution has lower regularity. In many applications, the solution of a second-order parabolic equation has only H 1 + s smoothness with 0 < s < 1 , and the numerical experiments show that the weak Galerkin approximate solution exhibits an optimal convergence order of 1 + s . However, the standard numerical analysis for weak Galerkin finite element method always requires that the exact solution should have at least H 2 smoothness. Our work fills the gap in the error analysis of weak Galerkin finite element method under lower regularity condition, where we prove the convergence order is of 1 + s . The main strategy of analysis is to introduce an H 2 -regular finite element approximation to discretize the spatial variables in variational equation, and then we analyze the error between this semi-discretized solution and the full discretized weak Galerkin solution. Finally, we present some numerical experiments to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fast linearized virtual element method (VEM) for the approximation of the nonlinear time-fractional diffusion equations on polygonal meshes. The L1-scheme with graded meshes is used to deal with the non-smooth system, the Newton linearized method is adopted to handle the nonlinear term and VEM is employed to discrete the spatial variable. Then the error splitting approach is used to prove the unconditional optimal error estimate of the fully discrete linearized L1-VEM scheme. In order to reduce the storage and computational cost caused by the nonlocality of the Caputo fractional operator, a fast memory-saving L1-VEM is developed. It is proved that the difference between the solution of the L1-VEM and the fast L1-VEM can be made arbitrarily small and is independently of the sizes of the time and/or space grids. Finally, numerical results are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a nonconforming extended virtual element method for solving elliptic interface problems with interface-unfitted meshes. The discrete approximation form is presented by adding some special terms along the edges of interface elements and several stabilization terms in the discrete bilinear form. The well-posedness of the discrete scheme is obtained and the optimal convergence is proven under the energy norm. It is shown that all results are independent of the position of the interface relative to the mesh and the contrast between the diffusion coefficients. Furthermore, short edges are allowed to appear in the mesh by modifying the stabilization term of the nonconforming virtual element method. Some numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In d dimensions, accurately approximating an arbitrary function oscillating with frequency ≲ k requires ∼ k d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k) suffers from the pollution effect if, as k → ∞ , the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, the hp-FEM (where accuracy is increased by decreasing the meshwidth h and increasing the polynomial degree p) does not suffer from the pollution effect. The heart of the proof of this result is a PDE result splitting the solution of the Helmholtz equation into “high” and “low” frequency components. This result for the constant-coefficient Helmholtz equation in full space (i.e. in ℝ d ) was originally proved in Melenk and Sauter (Math. Comp79(272), 1871–1914, 2010). In this paper, we prove this result using only integration by parts and elementary properties of the Fourier transform. The proof in this paper is motivated by the recent proof in Lafontaine et al. (Comp. Math. Appl.113, 59–69, 2022) of this splitting for the variable-coefficient Helmholtz equation in full space use the more-sophisticated tools of semiclassical pseudodifferential operators. [ABSTRACT FROM AUTHOR]

In the present paper the high-frequency induction welding process is studied numerically. The mathematical model comprises a harmonic vector potential formulation of the Maxwell equations and a quasi-static, convection dominated heat equation coupled through the Joule heat term and nonlinear constitutive relations. Its main novelties are a new analytic approach which permits to compute a spatially varying feed velocity depending on the angle of the Vee-opening and additional spring-back effects. Moreover, a numerical stabilization approach for the finite element discretization allows to consider realistic weld-line speeds and thus a fairly comprehensive three-dimensional simulation of the tube welding process. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze the conforming and nonconforming virtual element methods for the Signorini problem. Under some regularity assumptions, we prove optimal order a priori error estimates in the energy norm for both two numerical schemes. Extensive numerical tests are presented, verifying the theory and exploring unknown features. [ABSTRACT FROM AUTHOR]

One potential application of metamaterials is for designing invisibility cloaks. In this paper, we are interested in a rotation cloak model. Here we carry out the mathematical analysis of this model for the first time. Through a careful analysis, we reformulate a new system of governing partial differential equations by reducing one unknown variable from the originally developed modeling equations in Yang et al. (Commun Comput Phys 25:135–154, 2019). Then some novel finite element schemes are proposed and their stability and optimal error estimate are proved. Numerical simulations are presented to demonstrate that the new schemes for the reduced modeling equations can effectively reproduce the rotation cloaking phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze two stabilized finite element schemes for the Stokes–Darcy-transport model. The first stabilized method is a monolithic scheme with the backward-Euler time discretization, and fully coupled by one Stokes subproblem, one Darcy subproblem, and two transport subproblems. The second stabilized method is a non-iterative partitioned scheme which splits the fully coupled transport problem into two decoupled subproblems. The stability of the proposed schemes can be ensured by some strongly consistent interface terms. The numerical experiments are performed to illustrate the theoretical analysis and demonstrate the reliability and applicability of the proposed schemes. [ABSTRACT FROM AUTHOR]

The paper proposes a Petrov–Galerkin immersed finite element (PGIFE) method for solving Stokes interface problems. We employ the uniform mesh which is independent of interface. Immersed P 1 and P 0 spaces for velocity and pressure are constructed according to the jump conditions. We enrich the velocity space by the lowest RT0 element to ensure the inf-sup stability. The PGIFE method we developed yields the global divergence-free velocity and pressure-robustness. The existence and uniqueness of the approximated solution are derived. We obtain the optimal error estimate and prove that the error estimate for velocity is independent of pressure. Numerical experiments validate the optimal convergence results and the robustness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we design a multilevel local defect-correction method to solve the non-selfadjoint Steklov eigenvalue problems. Since the computation work needed for solving the non-selfadjoint Steklov eigenvalue problems increases exponentially as the scale of the problems increase, the main idea of our algorithm is to avoid solving large-scale equations especially large-scale Steklov eigenvalue problems directly. Firstly, we transform the non-selfadjoint Steklov eigenvalue problem into some symmetric boundary value problems defined in a multilevel finite element space sequence, and some small-scale non-selfadjoint Steklov eigenvalue problems defined in a low-dimensional auxiliary subspace. Next, the local defect-correction method is used to solve the symmetric boundary value problems, then the difficulty of solving these symmetric boundary value problems is further reduced by decomposing these large-scale problems into a series of small-scale subproblems. Overall, our algorithm can obtain the optimal error estimates with linear computational complexity, and the conclusions are proved by strict theoretical analysis which are different from the developed conclusions for equations with the Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect. [ABSTRACT FROM AUTHOR]

This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the L ∞ (L 2) -norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests. [ABSTRACT FROM AUTHOR]

This paper approximates viscosity solutions of fully nonlinear second order parabolic PDEs by a narrow-stencil finite difference spatial-discretization paired with the forward Euler method for the time discretization. Using generalized monotonicity of the numerical scheme and an iteration technique, we prove the numerical scheme is stable in both the l 2 -norm and the l ∞ -norm. Then under this stability, we establish the convergence of the proposed numerical scheme to the viscosity solution of the underlying fully nonlinear second order parabolic PDEs based on different regularities with respect to time and space. Finally, we report some numerical experiments. [ABSTRACT FROM AUTHOR]

Spatial network models are used as a simplified discrete representation in a wide range of applications, e.g., flow in blood vessels, elasticity of fiber based materials, and pore network models of porous materials. Nevertheless, the resulting linear systems are typically large and poorly conditioned and their numerical solution is challenging. This paper proposes a numerical homogenization technique for spatial network models which is based on the super-localized orthogonal decomposition (SLOD), recently introduced for elliptic multiscale partial differential equations. It provides accurate coarse solution spaces with approximation properties independent of the smoothness of the material data. A unique selling point of the SLOD is that it constructs an almost local basis of these coarse spaces, requiring less computations on the fine scale and achieving improved sparsity on the coarse scale compared to other state-of-the-art methods. We provide an a posteriori analysis of the proposed method and numerically confirm the method's unique localization properties. In addition, we show its applicability also for high-contrast channeled material data. [ABSTRACT FROM AUTHOR]

The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter C st , 1 that is found not robust as the polynomial degree p increases. This is related to the H 1 stability bound of the L 2 projection onto polynomials of degree at most p and its growth C st, 1 ∝ (p + 1) 1 / 2 as p → ∞ . A similar estimate for the Galerkin projection holds with a p-robust constant C st , 2 and C st , 2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant C st , 2 to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved L 2 error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates. [ABSTRACT FROM AUTHOR]

A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an H 2 conforming finite element space, an H (curl ; S) conforming finite element space, an H (div ; T) conforming finite element space and an L 2 finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element elasticity complex, a vector-valued H 1 conforming space and an H (curl curl T ; S) conforming space are constructed. Combining with an existing H (div ; S) ∩ H (div div ; S) element and an H (div ; S) element, respectively, these finite element spaces form two finite element elasticity complexes. The exactness of all the finite element complexes is proved. [ABSTRACT FROM AUTHOR]

This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. Since the Brinkman equations can be seen as a combination of the Stokes and Darcy equations, different conformities of finite element spaces are required depending on viscous parameters, making it challenging to develop a robust numerical solver uniformly performing for all viscous parameters. Therefore, we propose a pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method leads to error estimates independent of a pressure term and shows uniform performance for all viscous parameters, preserving minimal degrees of freedom. We prove well-posedness and error estimates for the robust EG method while comparing it with a standard EG method requiring an impractical mesh condition. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods’ performance to support the need for our robust method. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a class of globally divergence-free (and therefore pressure-robust) hybridizable discontinuous Galerkin (HDG) finite element methods for stationary Navier-Stokes equations. The methods use the P k / P k − 1 (k ⩾ 1) discontinuous finite element combination for the velocity and pressure approximations in the interior of elements, piecewise P m (m = k , k − 1) for the velocity gradient approximation in the interior of elements, and piecewise P k / P k for the trace approximations of the velocity and pressure on the inter-element boundaries. We show that the uniqueness condition for the discrete solution is guaranteed by that for the continuous solution together with a sufficiently small mesh size. Based on the derived discrete HDG Sobolev embedding properties, optimal error estimates are obtained. Numerical experiments are performed to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a quadratic B-spline collocation method is developed to solve a singularly perturbed semilinear reaction–diffusion problem with a discontinuous source term. The discontinuous source term leads to a jump in the second-order derivative of the exact solution at the discontinuous point. A quadratic B-spline collocation method on a Shishkin-type mesh is used to discretized the singularly perturbed problem on the left and right sides of the discontinuous point, respectively. The collocation equations at the discontinuous point are obtained using the conditions satisfied at the discontinuous point. It is shown that the scheme is stable and almost second-order uniformly convergent. Numerical experiments support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description, numerical simulation and theoretical analysis. Two important factors, heat and magnetic influences are involved. The mathematical model is formulated by four nonlinear partial differential equations (PDEs), determining four major physical variables. The influences of magnetic fields are supposed to be weak, and the strength is parallel to the z-axis. The elliptic equation is treated by a block-centered method, and the law of conservation is preserved. The computational accuracy is improved one order. Other equations are convection-dominated, thus are approximated by upwind block-centered differences. Upwind difference can eliminate numerical dispersion and nonphysical oscillation. The diffusion is approximated by the block-centered difference, while the convection term is treated by upwind approximation. Furthermore, the unknowns and adjoint functions are computed at the same time. These characters play important roles in numerical computations of conductor device problems. Using the theories of priori analysis such as energy estimates, the principle of duality and mathematical inductions, an optimal estimates result is obtained. Then a composite numerical method is shown for solving this problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a space-time finite element method for a time fractional diffusion interface problem. This method uses the low-order discontinuous Galerkin (DG) method and the Nitsche extended finite element method (Nitsche-XFEM) for temporal and spatial discretization, respectively. Sharp pointwise-in-time error estimates in graded temporal grids are derived, without any smoothness assumptions on the solution. Finally, three numerical examples are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we derive a novel recovery type a posteriori error estimation of the Crank–Nicolson finite element method for the Cahn–Hilliard equation. To achieve this, we employ both the elliptic reconstruction technique and a time reconstruction technique based on three time-level approximations, resulting in an optimal a posteriori error estimator. We propose a time-space adaptive algorithm that utilizes the derived a posteriori error estimator as error indicators. Numerical experiments are presented to validate the theoretical findings, including comparing with an adaptive finite element method based on a residual type a posteriori error estimator. [ABSTRACT FROM AUTHOR]

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients. [ABSTRACT FROM AUTHOR]

This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the L ∞ norm from the theory of viscosity solutions which are independent of the regularization parameter ε . They allow for the uniform convergence of the solution u ε to the regularized problem towards the Alexandrov solution u to the Monge–Ampère equation for any nonnegative L n right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the L ∞ norm for continuously differentiable finite element approximations of u or u ε . [ABSTRACT FROM AUTHOR]

Consider Laplace's equation in a bounded simply-connected domain S, and use the method of fundamental solutions (MFS). The error and stability analysis is made for circular/elliptic pseudo-boundaries in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020), and polynomial convergence rates and exponential growth rates of the condition number (Cond) are obtained. General pseudo-boundaries are suggested for more complicated solution domains in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020, Section 5). Since the ill-conditioning is severe, the success in computation by the MFS mainly depends on stability. This paper is devoted to stability analysis for smooth closed pseudo-boundaries of source nodes. Bounds of the Cond are derived, and exponential growth rates are also obtained. This paper is the first time to explore stability analysis of the MFS for non-circular/non-elliptic pseudo-boundaries. Circulant matrices are often employed for stability analysis of the MFS; but the stability analysis in this paper is explored based on new techniques without using circulant matrices as in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020). To pursue better pseudo-boundaries, the sensitivity index is proposed from growth/convergence rates of stability via accuracy. Better pseudo-boundaries in the MFS can be found by trial computations, to develop the study in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020) for the selection of pseudo-boundaries. For highly smooth and singular solutions, better pseudo-boundaries are different; an analysis of the sensitivity index is explored. Circular/elliptic pseudo-boundaries are optimal for highly smooth solutions, but not for singular solutions. In this paper, amoeba-like domains are chosen in computation. Several useful types of pseudo-boundaries are developed and their algorithms are simple without using nonlinear solutions. For singular solutions, numerical comparisons are made for different pseudo-boundaries via the sensitivity index. [ABSTRACT FROM AUTHOR]

This paper continues Part I (Hagstrom and Kim in Numer Math 141(4):917–966, 2019) of the investigation on the complete radiation boundary condition (CRBC) in waveguides. In this paper, we propose corner compatibility conditions for CRBC applied to the Helmholtz equation posed in R 2 . Since CRBC is developed as a high-order absorbing boundary condition approximating the radiation condition by using rational functions via the cross-sectional Fourier analysis, it is well-studied and its accurate performance is validated on a straight/planar fictitious boundary in waveguides. However in the presence of corners on artificial absorbing boundaries such as boundaries of rectangular domains, a special treatment for corner conditions is required. We design and validate the accurate CRBC with the corner compatibility conditions on rectangular domains. We also analyze the existence and uniqueness of solutions to the Helmholtz equation coupled with CRBC with the corner compatibility conditions. Finally, numerical experiments illustrating the accuracy of CRBC will be presented. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of immersed finite element (IFE) methods in three dimensions. Different from the 2D case, the points of intersection of the interface and the edges of a tetrahedron are usually not coplanar, which makes the extension of the original 2D IFE methods based on a piecewise linear approximation of the interface to the 3D case not straightforward. We address this coplanarity issue by an approach where the interface is approximated via discrete level set functions. This approach is very convenient from a computational point of view since in many practical applications the exact interface is often unknown, and only a discrete level set function is available. As this approach has also not be considered in the 2D IFE methods, in this paper we present a unified framework for both 2D and 3D cases. We consider an IFE method based on the traditional Crouzeix–Raviart element using integral values on faces as degrees of freedom. The novelty of the proposed IFE is the unisolvence of basis functions on arbitrary triangles/tetrahedrons without any angle restrictions even for anisotropic interface problems, which is advantageous over the IFE using nodal values as degrees of freedom. The optimal bounds for the IFE interpolation errors are proved on shape-regular triangulations. For the IFE method, optimal a priori error and condition number estimates are derived with constants independent of the location of the interface with respect to the unfitted mesh. The extension to anisotropic interface problems with tensor coefficients is also discussed. Numerical examples supporting the theoretical results are provided. [ABSTRACT FROM AUTHOR]

This paper presents the coupling method of direct discontinuous Garlekin (DDG) and natural boundary element (NBE) for exterior interface problem. The coupled DDG method is based on the Dirichlet transmission condition on a circular artificial boundary. The feature allows one to use continuous basis functions on the artificial boundary for the NBE method but apply discontinuous basis functions in interior domain bounded by this boundary for the DDG method and it is compatible with the spirit of DG methods. Based on trace, inverse estimate and optimal polynomial estimate for straight elements and curved elements caused by curved interface and circular artificial boundary, error and stability analysis allowed are presented. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyzing multigrid algorithm for solving elliptic interface problems discretized using the partially penalized immersed finite element (PPIFE). By taking the average values of nodal variables and integral variables, we construct intergrid transfer operators for the P 1 partially penalized immersed finite element ( P 1 -PPIFE) and the Crouzeix–Raviart partially penalized immersed finite element (CR-PPIFE), which satisfy certain stable approximation properties. An extra interface correction procedure is added in the smoothing steps to ensure the robustness of multigrid algorithm. We prove that the convergence of W-cycle multigrid algorithm and the condition number of the variable V-cycle as preconditioner are optimal by verifying the regularity-approximation assumption, which means that the convergence rate of algorithm is independent of mesh level, mesh size, and the position of the interface relative to the mesh. Numerical experiments illustrate the convergence of our algorithms using the W-cycle, V-cycle, and preconditioned conjugate gradient algorithm (PCG) with the V-cycle. [ABSTRACT FROM AUTHOR]

This paper introduces a high-order immersed finite element (IFE) method to solve two-phase incompressible Navier–Stokes equations on interface-unfitted meshes. In spatial discretization, we use the newly developed immersed P 2 - P 1 Taylor-Hood finite element. The unisolvency of new IFE basis functions is theoretically established. We introduce an enhanced partially penalized IFE method which includes the penalization on both interface edges and the interface itself. Ghost penalties are also added for pressure robustness. In temporal discretization, θ -schemes and backward differentiation formulas are adopted. Newton’s method is used to handle the nonlinear advection. The proposed method completely circumvent re-meshing in tackling moving-interface problems. Thanks to the isomorphism of our IFE spaces with the standard finite element spaces, the new method enables efficient updates of global matrices, which significantly reduces the overall computational cost. Comprehensive numerical experiments show that the proposed method is third-order convergent for velocity and second-order for pressure in both stationary and moving interface cases. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new structure-preserving finite element method for the stationary magnetohydrodynamic equations with magnetic-current formulation on Lipschitz domains. Using a mixed finite element approach, we discretize the hydrodynamic unknowns by inf-sup stable velocity-pressure finite element pairs, and the current density, the induced electric field and the magnetic field by using the edge-edge-face elements from a discrete de-Rham complex pair. To deal with the divergence-free condition of the magnetic field, we introduce an augmented term to the discrete scheme rather than a Lagrange multiplier in the existing schemes. Thanks to discrete differential forms and finite element exterior calculus, the proposed scheme preserves the divergence-free property exactly for the magnetic induction on the discrete level. The well-posedness of the discrete problem is further proved under the small data condition. Under weak regularity assumptions, we rigorously establish the error estimates of the finite element schemes. Numerical results are provided to illustrate the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, pressure projection stabilized low-order mixed finite element methods are studied to solve a Navier-Stokes hemivariational inequality for a boundary value problem of the Navier-Stokes equations involving a non-smooth non-monotone boundary condition. A new abstract mixed hemivariational inequality is introduced for the purpose of analyzing stabilized mixed finite element methods to solve the Navier-Stokes hemivariational inequality using velocity-pressure pairs without the discrete inf-sup condition. The well-posedness of the abstract problem is established through considerations of a related saddle-point formulation and fixed-point arguments. Then the results on the abstract problem are applied to the study of the Navier-Stokes hemivariational inequality and its stabilized mixed finite element approximations. Optimal order error estimates are derived for finite element solutions of the pressure projection stabilized lowest-order conforming pair and lowest equal order pair under appropriate solution regularity assumptions. Numerical results are reported on the performance of the pressure projection stabilized mixed finite element methods for solving the Navier-Stokes hemivariational inequality. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a virtual element method for the two dimensional elasticity spectral problem, where the polygonal meshes allow for the presence of small edges. Under this approach and with the aid of the theory of compact operators, we prove convergence for the proposed VEM and error estimates. We report a series of numerical tests in order to assess the performance of the method where we analyze the effects of the Poisson ratio on the computation of the order of convergence, together with the effects of the stabilization term on the arising of spurious eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, adaptive virtual element method (VEM) approximation of optimal control problem governed by Stokes equations with control constraints is discussed. The virtual element discrete scheme of the optimal control problem is constructed by polynomial projections and variational discretization of the control variable. Based on the a posteriori error estimates of VEM for Stokes equations and approximated error equivalence between the solutions of the optimal control problem and the solutions of the state and adjoint equations, we build up upper and lower bounds for the a posteriori error estimates of the optimal control problem. It proves that the a posteriori error indicator is reliable and efficient. The theoretical findings are illustrated by the numerical experiments. [ABSTRACT FROM AUTHOR]