A second-order face-centred finite volume strategy on general meshes is proposed. The method uses a mixed formulation in which a constant approximation of the unknown is computed on the faces of the mesh. Such information is then used to solve a set of problems, independent cell-by-cell, to retrieve the local values of the solution and its gradient. The main novelty of this approach is the introduction of a new basis function, utilised for the linear approximation of the primal variable in each cell. Contrary to the commonly used nodal basis, the proposed basis is suitable for computations on general meshes, including meshes with different cell types. The resulting approach provides second-order accuracy for the solution and first-order for its gradient, without the need of reconstruction procedures, is robust in the incompressible limit and insensitive to cell distortion and stretching. The second-order accuracy of the solution is exploited to devise an automatic mesh adaptivity strategy. An efficient error indicator is obtained from the computation of one extra local problem, independent cell-by-cell, and is used to drive mesh adaptivity. Numerical examples illustrating the approximation properties of the method and of the mesh adaptivity procedure are presented. The potential of the proposed method with automatic mesh adaptation is demonstrated in the context of microfluidics.
A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension
We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and compare the presented scheme to an entropy stable scheme based on simplicial summation-by-parts (SBP) finite difference operators. Finally, we report the computational performance of an implementa…
We devise a generalization of tree approximation that generates conforming meshes, i.e., meshes with a particular structure like edge-to-edge triangulations. A key feature of this generalization is that the choices of the cells to be subdivided are affected by that particular structure. As main result, we prove near best approximation with respect to conforming meshes, independent of constants like the completion constant for newest-vertex bisection. Numerical experiments complement the theoretical results and indicate better approximation properties than previous approaches.
Anisotropic mesh quality measures and anisotropic mesh adaptation are studied for polygonal meshes. Three sets of alignment and equidistribution measures are developed, one based on least squares fitting, one based on generalized barycentric mapping, and the other based on singular value decomposition of edge matrices. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given metrics. Based on one of these sets of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive poly…
The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (M…
A finite element solution of an ion channel dielectric continuum model such as Poisson-Boltzmann equation (PBE) and a system of Poisson-Nernst-Planck equations (PNP) requires tetrahedral meshes for an ion channel protein region, a membrane region, and an ionic solvent region as well as an interface fitted irregular tetrahedral mesh of a simulation box domain. However, generating these meshes is very difficult and highly technical due to the related three regions having very complex geometrical shapes. Currently, an ion channel mesh generation software package developed in Lu's research group i…
Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and devise numerical methods using locally refined meshes that lead to improved convergence rates despite the occurrence of discontinuities. It turns out that nearly linear convergence is possible on suitably constructed meshes.
In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical a…
A MATLAB program for computing differentiation matrices for arbitrary one-dimensional meshes is presented in this manuscript. The differentiation matrices for a mesh of N arbitrarily spaced points are formed from those obtained using Lagrangian interpolation on stencils of a fixed but arbitrary number M<=N of contiguous mesh points. For the particular case M=N and meshes with Chebyshev or Legendre distributions of points, the program yields the well known spectral differentiation matrices. For M<N and M odd, the differentiation matrices coincide, for the special case of an evenly spaced mesh, …
Adaptive mesh refinement is a key component of efficient unstructured space-time finite element methods. Underlying any adaptive mesh refinement scheme is, of course, a method for local refinement of simplices. However, simplex bisection algorithms in dimension greater than three have strict mesh preconditions which can be hard to satisfy. We prove that certain four-dimensional simplex space-time meshes can be handled with a relaxed precondition. Namely, we prove that if a tetrahedral mesh is 4-colorable, then we can produce a 4D simplex mesh which always satisfies the bisection precondition. …
The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the …
We consider a singularly perturbed convection-diffusion with exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost uniform convergence and superconvergence. The use of graded mesh allows for the SDFEM to prove almost uniform esimates in the SD norm, which is not possible for Shishkin type meshes.
Algebraically stabilized finite element discretizations of scalar steady-state convection-diffusion-reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is prov…
We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preser…
We design a Hybrid High-Order (HHO) scheme for the Poisson problem that is fully robust on polytopal meshes in the presence of small edges/faces. We state general assumptions on the stabilisation terms involved in the scheme, under which optimal error estimates (in discrete and continuous energy norms, as well as $L^2$-norm) are established with multiplicative constants that do not depend on the maximum number of faces in each element, or the relative size between an element and its faces. We illustrate the error estimates through numerical simulations in 2D and 3D on meshes designed by agglom…
Projecting fields between different meshes commonly arises in computational physics. This operation requires a supermesh construction and its computational cost is proportional to the number of cells of the supermesh $n$. Given any two quasi-uniform meshes of $n_A$ and $n_B$ cells respectively, we show under standard assumptions that n is proportional to $n_A + n_B$. This result substantially improves on the best currently available upper bound on $n$ and is fundamental for the analysis of algorithms that use supermeshes.
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives of the original function. The method generalises traditional finite difference methods to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy. We demonstrate the accuracy of the numerical scheme using dual quadrilateral meshes and a refinement method based on subdivision …
Finite Volume Element Method on Curved-Edge Meshes
School of Mathematics, Jilin University（吉林大学数学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所计算物理实验室）
This work introduces ``generalized meshes", a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend regular simplicial meshes by allowing for overlapping elements and more flexible adjacency relations. They can have several distinct ``generalized" vertices (or edges, faces) that occupy the same geometric position. These generalized facets are the natural degrees of freedom for classical conforming spaces of discrete differential forms appearing in finite and boundary element applications. Special attention is devoted…
This work concerns adaptive refinement procedures for meshes of polygonal virtual elements. Specifically, refinement procedures previously proposed by the authors for structured meshes are generalized for the challenging case of arbitrary element geometries arising in unstructured/Voronoi discretizations. Here, structured and unstructured meshes are considered and are created via Voronoi tessellation of sets of structured and unstructured seed points respectively. The novel mesh refinement procedures for both structured and unstructured meshes allow for accurate and efficient application of th…
A finite element cochain complex on Cartesian meshes of any dimension based on the H1-inner product is introduced. It yields H1-conforming finite element spaces with exterior derivatives in H1. We use a tensor product construction to obtain L2-stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.
This paper presents a lowest-order immersed Raviart-Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element finite (IFE) is constructed by modifying the traditional Raviart-Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal error estimates are rigorously proved for t…
High Order Finite Difference Methods on Non-uniform Meshes for Space Fractional Operators
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University（兰州大学数学与统计学院，甘肃省应用数学与复杂系统重点实验室）
We propose a new analysis of convergence for a $k$th order ($k\ge 1$) finite element method, which is applied on Bakhvalov-type meshes to a singularly perturbed two-point boundary value problem. A novel interpolant is introduced, which has a simple structure and is easy to generalize. By means of this interpolant, we prove an optimal order of uniform convergence with respect to the perturbation parameter. Numerical experiments illustrate these theoretical results.
A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based on modal energy coefficients is proposed to detect troubled cells. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted (NNW) interpolation is constructed to calculate troubled cells while smooth cells are calculated by the CPR method…
