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We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

The understanding of nonlinear, high dimensional flows, e.g, atmospheric and ocean flows, is critical to address the impacts of global climate change. Data Assimilation techniques combine physical models and observational data, often in a Bayesian framework, to predict the future state of the model and the uncertainty in this prediction. Inherent in these systems are noise (Gaussian and non-Gaussian), nonlinearity, and high dimensionality that pose challenges to making accurate predictions. To address these issues we investigate the use of both model and data dimension reduction based on techniques including Assimilation in Unstable Subspaces, Proper Orthogonal Decomposition, and Dynamic Mode Decomposition. Algorithms that take advantage of projected physical and data models may be combined with Data Analysis techniques such as Ensemble Kalman Filter and Particle Filter variants. The projected Data Assimilation techniques are developed for the optimal proposal particle filter and applied to the Lorenz'96 and Shallow Water Equations to test the efficacy of our techniques in high dimensional, nonlinear systems., 30 pages, 13 figures, 3 tables To appear in Computers & Mathematics with Applications, 2021,ISSN 0898-1221

An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper.

We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.

Sparse direct solvers using Block Low-Rank compression have been proven efficient to solve problems arising in many real-life applications. Improving those solvers is crucial for being able to 1) solve larger problems and 2) speed up computations. A main characteristic of a sparse direct solver using low-rank compression is when compression is performed. There are two distinct approaches: (1) all blocks are compressed before starting the factorization, which reduces the memory as much as possible, or (2) each block is compressed as late as possible, which usually leads to better speedup. The objective of this paper is to design a composite approach, to speedup computations while staying under a given memory limit. This should allow to solve large problems that cannot be solved with Approach 2 while reducing the execution time compared to Approach 1.We propose a memory-aware strategy where each block can be compressed either at the beginning or as late as possible. We first consider the problem of choosing when to compress each block, under the assumption that all information on blocks is perfectly known, i.e., memory requirement and execution time of a block when compressed or not. We show that this problem is a variant of the NP-complete Knapsack problem, and adapt an existing 2-approximation algorithm for our problem. Unfortunately, the required information on blocks depends on numerical properties and in practice cannot be known in advance. We thus introduce models to estimate those values. Experiments on the PaStiX solver demonstrate that our new approach can achieve an excellent trade-off between memory consumption and computational cost. For instance on matrix Geo1438, Approach 2 uses three times as much memory as Approach 1 while being three times faster. Our new approach leads to an execution time only 30% larger than Approach 2 when given a memory 30% larger than the one needed by Approach 1., Les solveurs directs creux utilisant de la compression low-rank ont montré leur efficacité pour résoudre de grands systèmes. Les améliorer permetde 1) résoudre de plus grands problèmes et 2) accélérer la résolution. Une caractéristique principale de ces solveurs est quand la compression est réalisée. Il existe deux approches: (1) tous les blocs sont compressés avant le début de la factorisation, ce qui minimise la consommation mémoire, ou (2) chaque bloc est compressé au plus tard, ce qui permet d’avoir de meilleures performances. L’objectif de cet article est de proposer une approche intermédiaire, qui accélère les calculs au maximum tout en restant en dessous d’une limite mémoire donnée. Cela devrait permettre de résoudre de grands problèmes que l’approche 2 ne peut pas résoudre ou de réduire le temps d’exécution de l’approche 1. Nous proposonsune stratégie memory-aware où chaque bloc peut ˆêtre compressé au début ou au plus tard. Nous commençons par considérer le problème où toutes les informations (consommation mémoire et temps d’exécution en version compressée ou non) sont parfaitement connues. Nous montrons que ce problème est une variante du problème NP-complet Knapsack et adaptons une 2-approximation pour notre problème. Malheureusement, toutes ces informations dépendent de propriétés numériques et ne sont pas connues `a l’avance. Des modèles sont donc introduits afin d’estimer ces valeurs. Des expérimentations avec le solveur PaStiX montrent que notre approche permet d’avoir un excellent compromis entre consommation mémoire et temps d’exécution. Par exemple, pour la matrice Geo1438, l’approche 2 utilise trois fois plus de mémoire que l’approche 1, qui est trois fois plus lente. Notre nouvelle méthode permet d’obtenir à la fois un temps d’exécution seulement 30% supérieur à celui de l’approche 2 tout en ayant une consommation mémoire seulement supérieure de 30% à celle de l’approche 1.

The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important.

The fractional Laplacian, ( − △ ) s , s ∈ ( 0 , 1 ) , appears in a wide range of physical systems, including Levy flights, some stochastic interfaces, and theoretical physics in connection to the problem of stability of the matter. In this paper, a matrix transfer technique (MTT) is employed combining with spectral/element method to solve fractional diffusion equations involving the fractional Laplacian. The convergence of the MTT method is analyzed by the abstract operator theory. Our method can be applied to solve various fractional equation involving fractional Laplacian on some complex domains. Numerical results indicate exponential convergence in the spatial discretization which is in good agreement with the theoretical analysis.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H 1 -norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The L p -norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.

Solving Burgers' equation always posses challenges for a small value of viscosity. Here we present a numerical method based on the Haar wavelet collocation method coupled with a nonstandard finite difference (NSFD) scheme for a class of generalized Burgers' equation. In the solution process, the time derivative is discretized by the NSFD scheme and the spatial derivatives are approximated by the Haar wavelets series. The nonlinear terms are linearized with the help of the quasilinearisation process. We illustrate the efficiency of the proposed method by solving several test problems and report their L 2 -error and L ∞ -error norms. The derived method is quite easy to implement compared to the other methods. Also, the error analysis of the current method is discussed. It is also observed that for the small number of grid points, the current method produces results that are in great agreement with the analytical solutions.

In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L 1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method.

In the context of time-domain simulation of integrated circuits, one often encounters large systems of coupled differential-algebraic equations. Simulation costs of these systems can become prohibitively large as the number of components keeps increasing. In an effort to reduce these simulation costs a twofold approach is presented in this paper. We combine maximum entropy snapshot sampling method and a nonlinear model order reduction technique, with multirate time integration. The obtained model order reduction basis is applied using the Gaus-Newton method with approximated tensors reduction. This reduction framework is then integrated using a coupled-slowest-first multirate integration scheme. The convergence of this combined method is verified numerically. Lastly it is shown that the new method results in a reduction of the computational effort without significant loss of accuracy.

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order q = p , some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L ∞ ( H 1 ) -norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

The Double Absorbing Boundary (DAB) is a recently proposed absorbing layer used to truncate an unbounded domain with high-order accuracy. While it was originally designed for time-dependent acoustics and elastodynamics, here the DAB construction is adapted and applied to the 2D Helmholtz equation. Both wave-guide and corner configurations are considered. A high-order spectral finite element scheme is used in order to match the discretization accuracy to the accuracy of the DAB. The DAB scheme is analyzed, and numerical experiments demonstrate its performance.

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω − . The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f, jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω − . The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

We give an effective method for solving fractional Riccati differential equations. We first define the fractional-order Boubaker wavelets (FOBW). Using the hypergeometric functions, we determine the exact values for the Riemann-Liouville fractional integral operator of the FOBW. The properties of FOBW, the exact formula, and the collocation method are used to transform the problem of solving fractional Riccati differential equations to the solution of a set of algebraic equations. These equations are solved via Newton's iterative method. The error estimation for the present method is also determined. The performance of the developed numerical schemes is assessed through several examples. This method yields very accurate results. The given numerical examples support this claim.

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$., 20 pages, 14 figures

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.

This paper is dedicated to solving the iterative solution to the discrete-time periodic Sylvester matrix equations. Inspired by Jacobi iterative algorithm and hierarchical identification principle, the Jacobi gradient based iterative (JGI) algorithm and the accelerated Jacobi gradient based iterative (AJGI) algorithm are proposed. It is verified that the proposed algorithms are convergent for any initial matrix when the parameter factor μ satisfies certain condition. The necessary and sufficient conditions are provided for the presented new algorithms. Moreover, we also apply the JGI algorithm and AJGI algorithm to study a more generalized discrete-time periodic matrix equations and give the convergence conditions of the algorithms. Finally, two numerical examples are given to illustrate the effectiveness, accuracy and superiority of the proposed algorithms.

An arbitrary order discontinuous Galerkin method in space and time is proposed to approximate the solution to hyperbolic systems of multi-dimensional conservation laws. Weak formulation is derived through the definition of weak divergence. Weak solution is given as a pair of weak functions on the element and the edge, respectively. Weak solution on the edge is characterized as the average of the solutions on the elements sharing the edge. Stability of the approximate solution is proved in a broken L 2 ( L 2 ) norm and also in a broken l ∞ ( L 2 ) norm. Error estimates of O ( h r + k n q ) with P r ( E ) and P q ( J n ) elements ( r , q > 1 + d 2 ) are then derived in a broken L 2 ( L 2 ) norm, where h and k n are the maximum diameters of the elements and the time step of J n , respectively, J n is the time interval, and d is the dimension of the spatial domain.

We consider terminal value problems for time-space fractional pseudo-parabolic equation subjected to a final/terminal value condition. In fact, fractional orders α , s are not exactly known in modeling. These are determined experimentally. The main purpose is to investigate the continuity of the solution with respect to the fractional order α ∈ ( 0 , 1 ) , which accordingly answer the question: does ρ α n → ρ α in an appropriate sense as α n → α ? Firstly, a formulation for integral solutions has been established, which based on Laplace transform and spectral expansion of the Mittag-Leffler operators. Then, the desired continuity will be obtained by making use of resolvent representations of the Mittag-Leffler operators on Hankel's contour. Finally, we present some numerical examples to illustrate the proposed theory.

This work proposes an approach to using the non-cooperative game theory for solving finite element problems. For this, the concept of generalized Nash equilibrium is applied to finite elements implying that each element is treated as a non-cooperative “player” in a larger finite element “game”. The aim of the approach is for all players to reach a Nash equilibrium ensuring that the entire discretization is at a minimum with respect to the decision variables considered. The approach is numerically demonstrated by investigating a nonlinear elasticity problem formulated as a finite element game. It is shown that the approach matches analytical solutions in linear elasticity and is convergent to a prescribed precision for two-player nonlinear problems.

A topology optimization method for steady-state flows of incompressible fluids which is capable of imposing accurate boundary conditions along the solid walls of the sought fluid paths is presented. In each topology optimization cycle, body-conforming Cartesian meshes are generated around shapes of any complexity by tracing the fluid-solid interfaces and a cut-cell flow solver is implemented together with its adjoint. Sensitivity derivatives are computed along the fluid-solid interfaces via the continuous adjoint method. Changes in topology are caused by expressing the computed sensitivity derivatives w.r.t. an auxiliary background material distribution, that helps updating the fluid-solid interfaces. The proposed method performance is assessed on three 2D benchmark examples and a 3D case. Two out of the three 2D examples are also solved using a porosity-based topology optimization approach in which impermeable regions are penalized by a Brinkman term and useful conclusions are drawn. For a fair comparison, designs optimized using the porosity-based method are re-evaluated after extracting fluid-solid interfaces from the computed porosity fields.

We present a variant of the immersed boundary method integrated with octree meshes for highly efficient and accurate Large-Eddy Simulations (LES) of flows around complex geometries. We demonstrate the scalability of the proposed method up to $\mathcal{O}(32K)$ processors. This is achieved by (a) rapid in-out tests; (b) adaptive quadrature for an accurate evaluation of forces; (c) tensorized evaluation during matrix assembly. We showcase this method on two non-trivial applications: accurately computing the drag coefficient of a sphere across Reynolds numbers $1-10^6$ encompassing the drag crisis regime; simulating flow features across a semi-truck for investigating the effect of platooning on efficiency., Comment: Accepted for publication at Computer and Mathematics with Applications

In this paper, we consider high order numerical methods for the solution of the initial-boundary value problem of two-sided space fractional advection-diffusion equations (SFADEs). We use the Crank-Nicolson (CN) technique to discretize the temporal derivative, and apply 3-point weighted and shifted Grunwald difference (WSGD) operators to discretize the fractional derivatives of order α ∈ ( 1 , 2 ) and fractional derivatives of order β ∈ ( 0 , 1 ) , respectively. As a result, a new family of CN-WSGD schemes for SFADEs with temporally 2nd order and spatially jth order ( j ≥ 2 ) accuracy are obtained. We then analyse the stability and the convergence of the numerical schemes. The extrapolated WSGD (EWSGD) operators are also discussed. Numerical examples are implemented to verify the theoretical results.

A meshless numerical scheme, named as space-time backward substitution method (STBSM), is proposed for 3D unsteady advection-diffusion equations (ADEs). In this method, the numerical solution is composed by approximation of boundary points and approximation of internal domain. Here, the space-time radial basis function (STRBF) is applied to construct internal approximation which satisfies homogeneous boundary condition. While, time dimension, as a new variable in radial coordinate, is incorporated into the global space-time basis function. Different from using other common global basis functions, the STBSM with STRBF does not require the temporal discretization, which makes it easy-to program and time saving. In addition, compactly supported space-time radial basis function (CS-STRBF) is introduced to reduce the original resultant full matrix to a sparse matrix, compared with global space-time radial basis function. It can overcome ill-conditioned matrix caused by an excessive condition number. Numerical results of two examples show that the proposed method provides accurate and stable solutions with low computational cost for three dimensional advection-diffusion equations. Moreover, comparison results indicate that the accuracy of the method is 2 − 4 orders of magnitude higher than the dimension splitting element-free Galerkin (DSEFG) [19] algorithm with less time expense.

This paper presents a topology optimization method for flow channel design using the lattice kinetic scheme (LKS). LKS is a numerical scheme merging the lattice Boltzmann method with the kinetic scheme for solving the incompressible Navier-Stokes equations. In the lattice Boltzmann method, the discrete distribution functions for every grid in the analysis domain need to be stored to compute the flow field, while in the LKS, only the flow velocities and the densities are stored, and thus requiring less storage. Moreover, in the LKS, the macroscopic boundary conditions in terms of the velocity and the pressure can be imposed directly rather than considering the bounce-back boundary condition against the distribution function. In this study, the optimization is performed based on the gradient of the objective functional with respect to the design variables and the design sensitivity is computed by the adjoint variable method. Both the forward and the adjoint problems are solved using the LKS. The validity of the design sensitivity is verified by comparing it with the finite difference approximation of objective functional. Numerical examples of solving two-dimensional flow channel design problems are presented to demonstrate the proposed method.

In this paper, the inverse problem of the determination of an unknown discontinuous parameter in elliptic equations is considered. A constrained optimization problem is formulated using the TV regularization and a specified selection term, in order to absorb the effect of noise and singularities. The discretization scheme used in this work is a combined finite element/volume method, then we establish the discrete version of optimization problem. The convergence and stability analysis are performed in term of noise level and diameter of singular set. We have considered the projected Wolfe algorithm to design our proposed minimization method. At the end we show some numerical experiments to prove the validity of theoretical results and the efficiency of the proposed method.

The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p ∈ [ 0 , 1 ] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples.

This paper presents a Cartesian grid based tailored finite point method (TFPM) for singularly perturbed reaction-diffusion equation on complex domains. The method is incorporated with the kernel-free boundary integral algorithm, where the semi-discrete boundary value problems after time integration are reformulated into corresponding Fredholm boundary integral equations (BIEs) of the second kind, however with no algorithmic dependence on the exact analytical expression for the kernels of integrals. The BIEs are iteratively solved by the GMRES method while integral evaluation during each iteration resorts to solving an equivalent interface problem, which in practice is achieved by a series of manipulations in the framework of TFPM including discretization, correction, solution, and interpolation. The proposed method has second-order accuracy for the reaction-diffusion equation as demonstrated by the numerical examples.

In this work, we studied the radial point interpolation collocation method (RIPCM) for the solution of the partial differential models with fractional derivatives. In the present meshless method, polynomial basis function and two novel types local support fields were considered. The (time-)space fractional differential equations with single-term or multi-term time derivatives in irregular domain were well simulated numerically by the proposed RIPCM method. Numerical tests were carried out to verify the accuracy and demonstrate the strength of the RIPCM in solving fractional derivative models in irregular domain.

Thermal tomography is a method for recovering heterogeneous thermal properties employing only boundary measurements. This paper focuses on the development of efficient inverse solvers for scenarios where the evolution of boundary conditions can vary in time. A transient heat model with two material parameters – volumetric capacity and a coefficient of thermal conductivity – is introduced for the description of the underlying physical phenomena. All proposed identification algorithms are deterministic methods based on a regularised Gauss-Newton method. A basic framework, implementation details, and the modification of general constraints initially derived for a standard setup of the Calderon problem are discussed here. Moreover, the algorithms are numerically verified for numerous examples, and results obtained show that the inverse problem exhibits a certain degree of ambiguity for a particular measurement-loading scenario. In other words, the important material property minimising the magnitude of error of the objective function seems to be the effusivity field rather than accurate identification of the individual thermal fields.

In this paper, a second order Gauge Uzawa scheme for the governing equations of an incompressible micropolar fluid flow is introduced and analyzed. The derivation of this scheme is obtained using the second order backward difference approximation. Later, the unconditional stability of the GUM scheme for the micropolar equations will be shown. Then, an a priori error estimate is established to prove the convergence. Finally, we present some numerical simulations that confirm the theoretical results.

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [26], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. Here, we describe the model together with its numerical discretization, and we prove a convergence result assuming the non-degeneracy of the phase mobilities and that the discrete solutions remain physical in the sense that, roughly speaking, the porosity does not vanish and the fractures remain open. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows in fractured porous media and the non-linear poromechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [36, 37]. Numerical tests employing the Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and $\mathbb P_2$ finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

In this paper, we introduce a multi-domain Muntz-polynomial spectral collocation method with graded meshes for solving second kind Volterra integral equations with a weakly singular kernel. This method is particularly suitable for problems whose solutions contain non-integer exponent factors. We carry out a rigorous error analysis of hp-version in the L ∞ - and weighted L 2 -norms. Several numerical examples are presented to demonstrate the efficiency and accuracy of the method.

In this manuscript, a novel modification of quintic B-spline differential quadrature method is given. By this modification, the obtained algebraic system do not have any ghost points. Therefore, the equation system is solvable and doesn't need any additional equation. Another advantage of this modification is the fact that the obtained numerical results are better than classical methods. Finite difference method is used as another component of the algorithm to obtain numerical solution of Korteweg-de Vries (KdV) equation which is the important model of nonlinear phenomena. Nine effective experiments namely single soliton, double solitons, interaction of two solitons, splitting to the solitons, interaction of three-, four-, and five solitons, evolution of solitons via Maxwellian initial condition and train of solitons are given with comparison of error norms and invariants. All of the given results with earlier studies show that the present scheme has improved classical common methods. Rate of convergence for single soliton and double solitons problems are reported.

In this paper, we study the numerical method for two dimensional fractional integro-differential equations, where the order of time fractional derivative α ∈ ( 1 , 2 ) and integral order γ ∈ ( 0 , 1 ) . To overcome the difficulty caused by the two fractional terms, we transform the original equation using the method of integration by parts. A novel high order compact alternating direction implicit (ADI) difference scheme is then proposed to solve the equivalent model. By some skills and detailed analysis, the unconditional stability and convergence in H 1 norm are proved, with the accuracy order O ( τ 2 + h 1 4 + h 2 4 ) , where τ , h 1 and h 2 are temporal and spatial step sizes, respectively. Finally, numerical results are presented to support the theoretical analysis.

We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The L 2 -projection method is implemented for the evaluation of numerical solutions by tracking the departure points from the integration points in each element. In the present study, a family of quadrature rules are used to enrich the approximation of integrals in the L 2 -projection method. The use of quadrature rules as an enrichment procedure allows for spatial discretizations on coarse fixed meshes and no need to introduce time-dependent enrichments. This procedure offers a very great advantage over the conventional Galerkin-characteristics finite element method since the same governing matrix representation can be used during the entire time steeping process. We also propose a multilevel adaptive procedure in the enriched Galerkin-characteristics finite element method by monitoring the gradient of the solution in the computational domain during its advection. In comparison with the traditional finite element analysis with h-, p- and hp-version refinements, the present approach is much simpler, more robust and efficient, and it yields more accurate solutions for a fixed number of degrees of freedom without refining the mesh. To examine the performance of the proposed method we solve several test examples for convection-diffusion problems. Comparison to the conventional Galerkin-characteristics finite element method is also carried out in the present work. The aim of such enriched method compared to the classical finite element method is to solve the time-dependent convection-dominated problems efficiently and with an appropriate level of accuracy.

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

An implementation of a massively parallel domain decomposing direct finite element equation solver named FALKSOL is tested using the parallel version of the state of the art open-source solver MUMPS (MUltifrontal Massively Parallel Solver) as a reference. FALKSOL includes its own very advanced multi-level domain decomposition and load balancing procedure. In the elemental mode, MUMPS uses a single level domain decomposition, and then solves the interface problem. FALKSOL offers a full out-of-core functionality, while MUMPS offers only a partial one. The problem size that can be solved by MUMPS is therefore hard limited by the amount of physical memory plus swap space 1 present. This is an actual bottleneck. FALKSOL can solve much larger problems. It is essentially only limited by the size of the secondary storage available. With modern so-called Non-Volatile Memory express (NVMe) on the PCI bus, enough I/O bandwidth is obtained with a couple of raided units. The results show that MUMPS is about twice as fast as FALKSOL in the limited range of problem size it copes with. The solvers are complementary. The out-of-core multi-level domain decomposition algorithm in FALKSOL makes it scale. The test indicates that there is a considerable potential gain in scalability/elasticity choosing the FALKSOL multi-level domain-decomposition out-of-core approach. Especially considering more powerful and larger compute systems. The price to be paid in speed compared with MUMPS is reasonable. The current stand-alone implementation of FALKSOL is interfaced with the 3Dhp code Demkowicz (2007) [32] , Demkowicz et al. (2008) [33] . The described MUMPS-like interface can be used to select the best solver for the task.

In this paper, we propose approximate solvers for two-phase fluids with general equations of state (EOS) in high dimension. The standard finite volume scheme is used for each fluid away from material interface. The two-phase interfaces are captured by the level set method coupled with a multi-resolution algorithm. For the Riemann solver of two-phase fluids with general equations of state, we construct an iterative approximation method by the solver for binomial equations of state. The velocity of the interface and the interface exchange fluxes are obtained precisely. With the help of the adaptive multi-resolution algorithms, we extend the method to three space dimensions conveniently. Numerical examples are carried out to demonstrate the strength and robustness of this method.

In this short paper, we introduce an efficient finite element method for solving the Stokes equations based on a divergence free H ( div ) element. The strong curl of C 0 - P k + 1 basis functions forms a basis of the divergence-free subspace of the P k - H ( div ) (i.e. B D M k ) finite element space. The [ P k + 1 ] 2 × 2 weak gradients are employed on these non- H 1 functions to obtain a conforming weak formulation. A strength of this approach is that the saddle point problem is replaced by a symmetric positive definite system with much less unknowns, which can be solved efficiently. Numerical examples are investigated to test accuracy and robustness of the method.

By employing the kernel functions with the form of piecewise polynomials in the Sobolev reproducing kernel Hilbert spaces (RKHSs), a globally superconvergent numerical technique is proposed to solve the second kind linear integral equations of Fredholm type. This method has an order of global convergence O ( h 4 ) and O ( h 6 ) based on the kernel functions in the Sobolev RKHSs H 1 and H 2 , respectively. Three linear Fredholm integral equations, one Volterra-Fredholm integral equation and one nonlinear Fredholm integral equation are numerically solved by the present approach to verify the superconvergence and effectiveness.

As a mesoscopic approach, the lattice Boltzmann method has achieved considerable success in simulating fluid flows and associated transport phenomena. The calculation, however, suffers from a massive amount of computing resources. A predictive model, to reduce the computing cost and accelerate the calculations, is proposed in this work. By employing an artificial neural network, composed of convolution layers and convolution long short-term memory layers, the model is an equivalent substitution of multiple time steps. A physical informed training loss function is introduced to improve the model predictive accuracy; and for the two-dimensional driven cavity problem, the mean square error of the prediction is less than 1.5 × 10 − 6 . For non-stationary flow, a time-dependent computing structure based on the current model is established. Nine iterative model calculations are performed consecutively for a two-dimensional driven cavity model, and the results are validated by comparing with the original (serial) lattice Boltzmann algorithm. Generally, in the case of training Reynolds number, for velocity and speed, the mean and the maximum absolute errors are lower than 0.012 and 0.12. Similarly, in the generalizing case, the mean and the maximum absolute errors are lower than 0.017 and 0.012. Besides, the current model's efficiency is about 15 times higher than that of the original lattice Boltzmann method.

In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrodinger equation by introducing a small regularized parameter 0 e ≪ 1 . The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrodinger equation. The regularized space-fractional logarithmic Schrodinger equation is proved to approximate the space-fractional logarithmic Schrodinger equation with linear convergence rate O ( e ) . The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrodinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrodinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrodinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.

In this article, novel linearized implicit difference schemes with energy conservation property for fractional Klein-Gordon-Zakharov system are constructed and analyzed. The important feature of the article is that new auxiliary equations ∂ u ∂ t = − v and ∂ ϕ ∂ t = ∂ 2 ψ ∂ x 2 are introduced to transform the original fractional Klein-Gordon-Zakharov system into an equivalent system of equations exactly. Especially, two kinds of efficacious difference operators, the leap-frog and modified Crank-Nicolson methods are respectively utilized to establish the linearized implicit difference schemes with energy conservation property for simulating the propagation of transformed equations. And above all, by employing the discrete energy method, we have proven that the constructed difference algorithms enjoy the convergence order of O ( Δ t 2 + h 2 ) and O ( Δ t 2 + h 4 ) in L ∞ - and L 2 -norms, without imposing any restrictive conditions on the grid ratio compared with the existing literature. Two numerical examples are carried out to investigate the physical behaviors of the wave propagation and substantiate the effectiveness of the suggested schemes.

In this paper, we explore the existence and uniqueness theorem for a problem of the fractional Cauchy form, with dependence on the generalized proportional Caputo derivative. Furthermore, a new numerical technique is presented based on a decomposition formula for the generalized proportional Caputo derivative. Convergence analysis of the proposed technique is proved. Finally, numerical results are obtained to confirm the validity of the proposed method.