Anisotropic Diffusion Problems Research Articles

The positivity‐preserving finite volume scheme with fixed stencils for anisotropic diffusion problems on general polyhedral meshes

AbstractTwo kinds of nonlinear cell‐centered positivity‐preserving finite volume schemes are proposed for the anisotropic diffusion problems on general three‐dimensional polyhedral meshes. First, the one‐sided flux on the cell‐faces is discretized using the fixed stencil of all vertices, then the cell‐centered discretization scheme is obtained using the nonlinear two‐point flux approximation. On this basis, a new explicit weighted second‐order vertex interpolation algorithm for arbitrary polyhedral meshes is designed to eliminate the vertex auxiliary unknowns in the scheme. In addition, an improved Anderson acceleration algorithm is adopted for nonlinear iteration. Finally, some benchmark examples are given to verify the convergence and positivity‐preserving property of the two schemes.

A posteriori element residual error estimations for the cell functional minimization scheme

This paper presents a posteriori element residual error estimator for the cell functional minimization scheme for anisotropic diffusion problems on unstructured polygonal meshes. The element residual error estimator is based on the numerical solution of the element residual equation on every cell. The global reliability and local efficiency of the proposed element residual error estimator with respect to an energy-type error are obtained.

Discrete strong extremum principles for finite element solutions of diffusion problems with nonlinear corrections

A nonlinear correction technique for finite element methods to anisotropic diffusion problems on general triangular and quadrilateral meshes are introduced. The classic linear or bi-linear finite element methods are modified, and then the resulting schemes satisfy the discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well-known restrictions on diffusion coefficients and geometry of mesh-cell (e.g. “acute angle” condition). Convergence rate for smooth and piecewise smooth solutions and the discrete extremum principle property are verified by numerical examples.

A least squares based diamond scheme for 3D heterogeneous and anisotropic diffusion problems on polyhedral meshes

We propose a diamond scheme for 3D heterogeneous and anisotropic problem using piecewise linear approximation and least squares method. Since the continuity of the solution and flux is treated appropriately, the scheme and the vertex interpolation algorithm allow arbitrary diffusion tensors. In addition, we employ an adaptive weighted coefficient in the least squares problem of the vertex interpolation using a novel way to evaluate the magnitude of the diffusion anisotropy. Consequently our vertex interpolation has better performance than the least squares interpolation in most tested cases. The presented scheme and vertex interpolation algorithm do not need edge information, which leads to less topological searches and simplifies the programming compared with other edge-based ones. By means of unified calculations of the geometric quantities, this scheme is suitable for meshes with nonplanar faces. Numerical experiments show that our scheme is linearity-preserving and achieves nearly optimal accuracy for the solution on general meshes. Moreover, the new vertex interpolation algorithm has an ideal performance for heterogeneous and highly anisotropic diffusion problems which are tough for some existing algorithms.

Performances of Hybridized-, Embedded-, and Weighted-Interior Penalty Discontinuous Galerkin Methods for Heterogeneous and Anisotropic Diffusion Problems

The present paper discusses families of Interior Penalty Discontinuous Galerkin (IP) methods for solving heterogeneous and anisotropic diffusion problems. Specifically, we focus on distinctive schemes, namely the Hybridized-, Embedded-, and Weighted-IP schemes, leading to final matrixes of different sizes and sparsities. Both the Hybridized- and Embedded-IP schemes are eligible for static condensation, and their degrees of freedom are distributed on the mesh skeleton. In contrast, the unknowns are located inside the mesh elements for the Weighted-IP variant. For a given mesh, it is well-known that the number of degrees of freedom related to the standard Discontinuous Galerkin methods increases more rapidly than those of the skeletal approaches (Hybridized- and Embedded-IP). We then quantify the impact of the static condensation procedure on the computational performances of the different IP classes in terms of robustness, accuracy, and CPU time. To this aim, numerical experiments are investigated by considering strong heterogeneities and anisotropies. We analyze the fixed error tolerance versus the run time and mesh size to guide our performance criterion. We also outlined some relationships between these Interior Penalty schemes. Eventually, we confirm the superiority of the Hybridized- and Embedded-IP schemes, regardless of the mesh, the polynomial degree, and the physical properties (homogeneous, heterogeneous, and/or anisotropic).

Monotonicity correction for second order element finite volume methods of anisotropic diffusion problems

We apply the monotonicity correction to second-order element finite volume methods, which include quadratic element on triangular meshes and biquadratic element on quadrilateral meshes, for anisotropic diffusion problems, and obtain a second-order monotone finite volume scheme. The main ideas of monotonicity correction are as follows: When we formulate the second-order element finite volume schemes, we need to calculate line integrals on the boundary line segment of the dual element, and regard these line integrals as numerical fluxes, which are in the form of multi-point flux. Among these numerical fluxes, we can separate a two-point flux structure on both sides of the line segment. Therefore, we can apply nonlinear monotone correction to these numerical fluxes, then we obtain a corrected second order element finite volume schemes whose stiffness matrix is monotone matrix, so the first high order unconditional positivity-preserving finite volume schemes are obtained by us. Moreover, we analyze the truncation error of the corrected numerical fluxes. Furthermore, for the time dependent problem, we also apply monotone correction to time derivative term. Numerical experiments show that the corrected second order element finite volume schemes have monotonicity, and maintain the convergence order of the original schemes, which are second order in H1 norm and third order in L2 norm.

A nonlinear correction scheme for the heterogeneous and anisotropic diffusion problems on polygonal meshes

In this paper, a nonlinear positivity-preserving finite volume scheme for the heterogeneous and anisotropic diffusion problems is proposed. Firstly, a linear diamond finite volume scheme is introduced. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as auxiliary ones and are expressed as linear combinations of the surrounding cell-centered unknowns, which reduces the scheme to a pure cell-centered one. Two new interpolation algorithms are suggested through the linearity-preserving approach and a novel discretization of diffusion coefficient. The new vertex interpolation algorithms improve a great deal the numerical performance of the linear scheme on distorted meshes with strongly anisotropy. Secondly, a new positivity-preserving cell-centered scheme is constructed via the aforementioned linear scheme and a nonlinear correction technique proposed by [28]. This nonlinear scheme is different from most existing nonlinear two-point flux approximation schemes in the way that it is not based on the nonlinear two-point flux approximation and it doesn't require the positivity-preserving vertex interpolation. Its implementation is also very easy and simple. Numerical experiments conducted in this paper indicate that the new positivity-preserving scheme is efficient and has approximately second order accuracy in most extreme cases.

A family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes

In this paper, we propose and analyze a family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes. The present work covers a previous one Zhou and Wu (2020) where only the case of scalar diffusion coefficients is considered. The novelty of this paper is that, by the introduction of some special parameters involving the diffusion tensor and the mesh geometry, each entry of the element stiffness matrix can be split as three parts that are easily computed and analyzed. Thanks to this representation, we manage to obtain a sufficient condition that ensures the existence, uniqueness and coercivity result of the finite volume element solution on triangular mesh. Moreover, this sufficient condition has a simple, analytic and computable expression that can be easily judged on any triangular meshes with arbitrary full diffusion tensors. In particular, when the diffusion tensor is an identity matrix, this condition reduces to a geometric one and only relies on the interior angles of each triangular element, which is consistent with the result in (Zhou and Wu, 2020). Moreover, based on this sufficient condition, we obtain a minimum angle condition for the case of general diffusion tensors, which is better than some existing ones. By the coercivity result, the optimal H1 error estimate is also obtained. Finally, the theoretical results are verified by some numerical experiments.

Discrete Optimization of Robin Transmission Conditions for Anisotropic Diffusion with Discrete Duality Finite Volume Methods

Discrete Duality Finite Volume (DDFV) methods are very well suited to discretize anisotropic diffusion problems, even on meshes with low mesh quality. Their performance stems from an accurate reconstruction of the gradients between mesh cell boundaries, which comes however at the cost of using both a primal (cell centered) and a dual (vertex centered) mesh, and thus leads to larger system sizes. To solve these systems, we propose to use non-overlapping optimized Schwarz methods with Robin transmission conditions, which can also well take into account anisotropic diffusion across subdomain interfaces. We study these methods here directly at the discrete level, and prove convergence using energy estimates for general decompositions including cross points and fully anisotropic diffusion. Our analysis reveals that primal and dual meshes might be coupled using different optimized Robin parameters in the optimized Schwarz methods. We present both the separate and coupled optimization of Robin transmission conditions and derive parameters which lead to the fastest possible convergence in each case. We illustrate our results with numerical experiments for the model problem, and also in situations that go beyond our analysis, with an application to anisotropic image reconstruction.

A least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes

AbstractWe present a new least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes. This scheme introduces both cell‐centered unknowns and vertex unknowns. The vertex unknowns are intermediate ones and are expressed as linear combinations with the surrounding cell‐centered unknowns by a new vertex interpolation algorithm which is also derived in least squares approach. Both the new scheme and the vertex interpolation algorithm are applicable to diffusion problems with arbitrary diffusion tensors and do not depend on the location of discontinuity. Benefitting from the flexibility of least squares approach, the new scheme and vertex interpolation algorithm can also be extended to 3D cases naturally. The new scheme and vertex interpolation algorithm are linearity‐preserving under given assumptions and the numerical results show that they maintain nearly optimal convergence rates for both error and error in general cases. More interesting is that a very robust performance of the new vertex interpolation algorithm on random meshes compared with the algorithm LPEW2 is found from the numerical tests.

The cell-centered positivity-preserving finite volume scheme for 3D anisotropic diffusion problems on distorted meshes

In this paper, a new cell-centered positivity-preserving finite volume scheme is proposed for the 3D anisotropic diffusion problems on distorted meshes. The primary unknowns and auxiliary unknowns are used in this scheme. First, the one-sided flux on each cell facet is discretized by a fixed stencil. Then, the nonlinear two-point flux approximation is applied to get the cell-centered discrete scheme. Besides, a second-order interpolation algorithm is constructed to eliminate the auxiliary unknowns in flux expressions. Furthermore, an improved Anderson acceleration method is given to accelerate the convergence of Picard iterations. Finally, several numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

High order locally conservative finite element solutions for anisotropic diffusion problems in two dimensions

For the two dimensional anisotropic diffusion problems, we postprocess the k-th order finite element solution to satisfy the local conservation law on a certain dual mesh. The postprocessing technique is independently implemented by solving a 3-by-3 and 4-by-4 local linear algebraic system on each triangular and quadrilateral element, respectively. For any full anisotropic diffusion tensors, and arbitrary triangular and trapezoidal meshes, the existence and uniqueness of the postprocessed solutions are proved theoretically. Consequently, some existing results are improved, and the traditional h1+γ-parallelogram assumption is replaced with a more general one. The optimal H1 and L2 norm error estimates of the postprocessed solutions are also proved. Finally, some numerical experiments are presented to validate the theoretical findings.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Abstract We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

Tailored finite point method for the approximation of diffusion operators with non-symmetric diffusion tensor

We present a tailored finite point method (TFPM) for anisotropic diffusion equations with a non-symmetric diffusion tensor on Cartesian grids. The fluxes on each edge are discretized by using a linear combination of the local basis functions, which come from the exact solution of the diffusion equation with constant coefficients on the local cell. In this way, the scheme is fully consistent and the flux is naturally continuous across the interfaces between the subdomains with a non-symmetric diffusion tensor. Additionally, it is convenient to handle the Neumann boundary condition or a variant of Neumann boundary condition. Numerical results obtained from solving different anisotropic diffusion problems including problems with sharp discontinuity near the interface and boundary, show that this approach is efficient. Second order convergence rate can be obtained with the numerical examples. The new scheme is also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics, the results further show the robustness of this new method.

A high-order non field-aligned approach for the discretization of strongly anisotropic diffusion operators in magnetic fusion

In this work we present a hybrid discontinuous Galerkin scheme for the solution of extremely anisotropic diffusion problems arising in magnetized plasmas for fusion applications. Unstructured meshes, non-aligned with respect to the dominant diffusion direction, allow an unequalled flexibility in discretizing geometries of any shape, but may lead to spurious numerical diffusion. Curved triangles or quadrangles are used to discretize the poloidal plane of the machine, while a structured discretization is used in the toroidal direction. The proper design of the numerical fluxes guarantees the correct convergence order at any anisotropy level. Computations performed on well-designed 2D and 3D numerical tests show that non-aligned discretizations are able to provide spurious diffusion free solutions as long as high-order interpolations are used. Introducing an explicit measure of the numerical diffusion, a careful investigation is carried out showing an exponential increase of this latest with respect to the non-alignment of the mesh with the diffusion direction, as well as an exponential decrease with the polynomial degree of interpolation. A brief assessment of the method with respect to two finite-difference schemes using non-aligned discretization, but classically used in fusion modeling, is also presented. Program summaryProgram Title: Laplace-HDG (Laplace Hybrid Discontinuous Galerkin)CPC Library link to program files:http://dx.doi.org/10.17632/c3dhycyvj8.1Licensing provisions: GPLv3Programming language: Fortran 95Nature of problem: Anisotropic Laplace problem in 2D with Dirichlet boundary conditionsSolution method: Hybrid discontinuous Galerkin scheme

Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique

<p style='text-indent:20px;'>We present in this paper a new algorithm combining a finite volume method with an improved Schur complement technique to solve <inline-formula><tex-math id="M1">\begin{document}$ 2D $\end{document}</tex-math></inline-formula> anisotropic diffusion problems on general meshes. After having proved the convergence of the finite volume method, we have given a description of the proposed algorithm in the case of two nonoverlapping subdomains. Several numerical tests are achieved which illustrate the theoretical results of convergence of the finite volume method and show the advantages of the proposed algorithm.

A novel cell-centered finite volume scheme with positivity-preserving property for the anisotropic diffusion problems on general polyhedral meshes

In this paper, we propose a novel cell-centered positivity-preserving finite volume scheme for the anisotropic diffusion problems. The discretization of diffusion flux is based on the standard nonlinear two-point flux approximation. The cell vertexes are employed to define auxiliary unknowns. A positivity-preserving vertex interpolation algorithm is constructed to get the value of auxiliary unknowns. This interpolation algorithm has almost second order convergence rate on the distorted meshes. Numerical results illustrate that our scheme is efficient and practical in solving the 3D anisotropic diffusion problems.

A [formula omitted]-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh

In this paper, we study a Q1-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh. It is known that the coercivity is the basement for some other theoretical results (stability, H1 and L2 error estimates, etc.) and the existing results were mainly obtained on h1+γ-parallelogram meshes with scalar diffusion coefficients. For the cases of full diffusion tensors and arbitrary convex quadrilateral meshes, we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme. More interesting is that, this sufficient condition covers the traditional h1+γ-parallelogram mesh assumption and has an explicit expression, by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h>0. Moreover, an H1 error estimate is obtained without the h1+γ-parallelogram assumption, and some numerical results are also provided to validate the theoretical results.

Optimized Schwarz and finite element cell-centered method for heterogeneous anisotropic diffusion problems

The paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters.

A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes

We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.