A nine-point scheme with explicit weights for diffusion equations on distorted meshes

In this paper, for the structured quadrilateral mesh we derive a nine-point difference scheme which has five cell-centered unknowns and four vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear combination of the neighboring cell-centered unknowns, which reduces the scheme to a cell-centered one with a local stencil involving nine cell-centered unknowns. The coefficients in the linear combination are known as the weights and two types of new weights are proposed. These new weights are neither discontinuity dependent nor mesh topology dependent, have explicit expressions, can reduce to the one-dimensional harmonic-average weights on the nonuniform rectangular meshes, and moreover, are easily extended to the unstructured polygonal meshes and non-matching meshes. Both the derivation of the nine-point scheme and that of new weights satisfy the linearity preserving criterion. Numerical experiments show that, with these new weights, the nine-point difference scheme and its simple extension have a nearly second order accuracy on many highly distorted meshes, including structured quadrilateral meshes, unstructured polygonal meshes and non-matching meshes.     

三维多面体网格上扩散方程的保正格式

 针对三维任意(星形)多面体网格, 本文构造了扩散方程的一种单元中心型非线性有限体积格式, 证明了该格式具有保正性. 在该格式设计中, 除引入网格中心量外, 还引入网格节点量和网格面中心量作为中间未知量, 它们将用网格中心未知量线性组合表示, 使得格式仅有网格中心未知量作为基本未知量. 在节点量计算中, 利用网格面上的调和平均点, 设计了一种适用于三维多面体网格的局部显式加权方法. 该格式适用于求解非平面的网格表面和间断扩散系数的问题. 数值例子验证了它对光滑解具有二阶精度和保正性.     

An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes

In this paper, we suggest a new vertex interpolation algorithm to improve an existing cell-centered finite volume scheme for nonlinear diffusion problems on general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability of the corresponding local system is proved under some assumptions. Additionally, we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The final scheme allows arbitrary continuous or discontinuous diffusion coefficients and can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux are observed in numerical experiments. In the comparative experiments with some existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.     

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes. The vertex unknowns are treated as primary ones for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is not convex in general, leading to a fixed stencil of the flux approximation and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.     

扩散方程保正的有限体积格式

在大变形网格上数值求解多介质扩散方程时, 如何构造具有保正性的扩散格式一直是人们关注的难题. 本文将简要综述与保正性相关的扩散格式的研究历史, 并为解决这一难题提出新的设计途径,构造出新的具有较高精度的单元中心型守恒保正格式, 它们可兼顾网格几何变形和物理量变化. 本文将给出数值实验结果, 验证新格式在变形的网格上保持非负性.     

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.     

多边形网格上扩散方程新的单调格式

 本文在星形多边形网格上, 构造了扩散方程新的单调有限体积格式.该格式与现有的基于非线性两点流的单调格式的主要区别是, 在网格边的法向流离散模板中包含当前边上的点, 在推导离散法向流的表达式时采用了定义于当前边上的辅助未知量, 这样既可适应网格几何大变形, 同时又兼顾了当前网格边上物理量的变化. 在光滑解情形证明了离散法向流的相容性.对于具有强各向异性、非均匀张量扩散系数的扩散方程, 证明了新格式是单调的, 即格式可以保持解析解的正性. 数值结果表明在扭曲网格上, 所构造的格式是局部守恒和保正的, 对光滑解有高于一阶的精度, 并且, 针对非平衡辐射限流扩散问题, 数值结果验证了新格式在计算效率和守恒精度上优于九点格式.     

A new multipoint flux approximation method with a quasi-local stencil (MPFA-QL) for the simulation of diffusion problems in anisotropic and heterogeneous media

In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature.     

Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

The focal geometry of circular and conical meshes

Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivision-like refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a two-parameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the three-dimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.      

A virtual element method-based positivity-preserving conservative scheme for convection–diffusion problems on polygonal meshes

In this paper, we propose a positivity-preserving conservative scheme based on the virtual element method (VEM) to solve convection–diffusion problems on general meshes. As an extension of finite element methods to general polygonal elements, the VEM has many advantages such as substantial mathematical foundations, simplicity in implementation. However, it is neither positivity-preserving nor locally conservative. The purpose of this article is to develop a new scheme, which has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids. The first step is to calculate the cell-vertex values by the lowest-order VEM. Then, the nonlinear two-point flux approximations are utilized to obtain the nonnegativity of cell-centered values and the local conservation property. The new scheme inherits both advantages of the VEM and the nonlinear two-point flux approximations. Numerical results show that the new scheme can reach the optimal convergence order of the virtual element theory, that is, the second-order accuracy for the solution and the first-order accuracy for its gradient. Moreover, the obtained cell-centered values are nonnegative, which demonstrates the positivity-preserving property of our new scheme.     

Graded Meshes for Higher Order FEM

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an 'optimal' mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.     

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.     

On the adjacencies of triangular meshes based on skeleton-regular partitions

For any 2D triangulation τ, the 1-skeleton mesh of τ is the wireframe mesh defined by the edges of τ, while that for any 3D triangulation τ, the 1-skeleton and the 2-skeleton meshes, respectively, correspond to the wireframe mesh formed by the edges of τ and the “surface” mesh defined by the triangular faces of τ. A skeleton-regular partition of a triangle or a tetrahedra, is a partition that globally applied over each element of a conforming mesh (where the intersection of adjacent elements is a vertex or a common face, or a common edge) produce both a refined conforming mesh and refined and conforming skeleton meshes. Such a partition divides all the edges (and all the faces) of an individual element in the same number of edges (faces). We prove that sequences of meshes constructed by applying a skeleton-regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements, faces, edges and vertices of the nth and (n−1)th meshes. By using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes (number of triangles by node and number of tetrahedra by vertex) is constant when n goes to infinity. We relate these results with the non-degeneracy properties of longest-edge based partitions in 2D and include empirical results which support the conjecture that analogous results hold in 3D.     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

Anisotropic $EQ_1^{rot}$ Finite Element Approximation for a Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation

By employing $EQ_1^{rot}$ nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of $EQ_1^{rot}$ nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.     

Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations

We present a new algorithm for generating layer-adapted meshes for the finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar to the well-known Bakhvalov meshes, but can be used in more general settings: specifically two-dimensional problems for which the optimal mesh is not tensor-product in nature. Our focus is on the efficient implementation of these algorithms, and numerical verification of their properties in a variety of settings. The MPDE is a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh intervals. We resolve this by proposing a scheme based on $h$-refinement. We present fully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)] that implement these methods, facilitating their extension to other problems and settings.     

A POSTERIORI ERROR ESTIMATION OF THE NEW MIXED ELEMENT SCHEMES FOR SECOND ORDER ELLIPTIC PROBLEM ON ANISOTROPIC MESHES

This paper presents a posteriori residual error estimator for the new mixed el-ement scheme for second order elliptic problem on anisotropic meshes. The reliability and efficiency of our estimator are established without any regularity assumption on the mesh.     

A posteriori error estimation of the new mixed element schemes for second order elliptic problem on anisotropic meshes

This paper presents a posteriori residual error estimator for the new mixed element scheme for second order elliptic problem on anisotropic meshes. The reliability and efficiency of our estimator are established without any regularity assumption on the mesh.