Analysis and construction of cell-centered finite volume scheme for diffusion equations on distorted meshes

A finite volume scheme solving diffusion equation on non-rectangular meshes is introduced by Li [Deyuan Li, Hongshou Shui, Minjun Tang, On the finite difference scheme of two-dimensional parabolic equation in a non-rectangular mesh, J. Numer. Meth. Comput. Appl. 4 (1980) 217 (in Chinese), D.Y. Li, G.N. Chen, An Introduction to the Difference Methods for Parabolic Equation, Science Press, Beijing, 1995 (in Chinese)], which is the so-called nine-point scheme on arbitrary quadrangles. The vertex unknowns can be represented as some weighted combination of the cell-centered unknowns, but it is difficult to choose the suitable combination coefficients for the multimaterial computation on highly distorted meshes. We present a nine-point scheme for discretizing diffusion operators on distorted quadrilateral meshes, and derive a new expression for vertex unknowns. The stability and convergence of the resulting scheme are proved. We give numerical results for various test cases which exhibit the good behavior of our scheme.     

非结构任意多边形网格辐射扩散方程有限体积格式

本文基于非结构任意多边形网格体系,给出了求解辐射扩散方程的中心型有限体积格式,格式中出现的网格节点未知量由相邻的网格中心未知量加权给出,综合考虑网格几何及扩散系数的影响,给出了节点未知量的一种加权方式,数值实验表明格式在各种非结构网格上具有较强的适应性.     

Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

We study in this paper a posteriori error estimates for H ¹-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.     

Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the \(L^\infty \)-norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.     

A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes

In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation converges to the exact solution with the optimal rate in the energy norm. Furthermore, we discuss superconvergence property of the FVM solution. With the help of this superconvergence result, we find that the FVM solution also converges to the exact solution with the optimal rate in the $L^2$ -norm. Finally, we validate our theory with numerical experiments.     

Numerical identification of a nonlinear diffusion coefficient by discrete mollification

The discrete mollification method is a convolution-based filtering procedure suitable for the regularization of ill-posed problems. Combined with explicit space-marching finite difference schemes, it provides stability and convergence for a variety of coefficient identification problems in linear parabolic equations. In this paper, we extend such a technique to identify some nonlinear diffusion coefficients depending on an unknown space dependent function in one dimensional parabolic models. For the coefficient recovery process, we present detailed error estimates and to illustrate the performance of the algorithms, several numerical examples are included.     

A generalization of the vertex-centered finite volume scheme to arbitrary high order

A higher order finite volume method for elliptic problems is proposed for arbitrary order ${p \in \mathbb{N}}$ . Piecewise polynomial basis functions are used as trial functions while the control volumes are constructed by a vertex-centered technique. The discretization is tested on numerical examples utilizing triangles and quadrilaterals in 2D. In these tests the optimal error is achieved in the H ¹-norm. The error in the L ²-norm is one order below optimal for even polynomial degrees and optimal for odd degrees.     

Inflow-Based Gradient Finite Volume Method for a Propagation in a Normal Direction in a Polyhedron Mesh

An inflow-based gradient is proposed to solve a propagation in a normal direction with a cell-centered finite volume method. The proposed discretization of the magnitude of gradient is an extension of Rouy–Tourin scheme (SIAM J Numer Anal 29:867–884, 1992) and Osher–Sethian scheme (J Comput Phys 79:12–49, 1988) in two cases; the first is that the proposed scheme can be applied in a polyhedron mesh in three dimensions and the second is that its corresponding form on a regular structured cube mesh uses the second order upwind difference. Considering a practical application in three dimensional mesh, we use the simplest decomposed domains for a parallel computation. Moreover, the implementation is straightforwardly and easily combined with a conventional finite volume code. A higher order of convergence and a recovery of signed distance function from a sparse data are illustrated in numerical examples on hexahedron or polyhedron meshes.     

Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by the nonlinear degenerate parabolic system

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge–Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.     

A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations

A hybrid staggered discontinuous Galerkin method is developed for the Korteweg–de Vries equation. The equation is written into a system of first order equations by introducing auxiliary variables. Two sets of finite element functions are introduced to approximate the solution and the auxiliary variables. The staggered continuity of the two finite element function spaces gives a natural flux condition and trace value on the element boundaries in the derivation of Galerkin approximation. On the other hand, to deal with the third order derivative term an hybridization idea is used and additional flux unknowns are introduced. The auxiliary variables can be eliminated in each element and the resulting algebraic system on the solution and the additional flux unknowns is solved. Stability of the semi discrete form is proven for various boundary conditions. Numerical results present the optimal order of \(L^2\)-errors of the proposed method for a given polynomial order.     

Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

In this paper, we present unconditionally optimal error estimates of linearized Crank–Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in \(\mathbb {R}^d\ (d=2,3)\). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal \(L^2\) error estimates of the numerical schemes can be obtained under the condition \(\tau \ge h\), where \(\tau \) denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when \(\tau \le h\). Then, optimal \(L^2\) and \(H^1\) error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.     

Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation,stability, and error analysis

Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat). The approximation at the grid points near the fine and coarse grid interface is based on the approach proposed by the authors in a previous paper for selfadjoint elliptic equations. The proposed schemes are implicit of backward Euler type and are shown to be unconditionally stable. Error analysis is also presented.     

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete \(L^{\infty }(L^2)\) and \(L^2(H^1)\) errors are \(O(\triangle t^{2-\alpha }+h^2+H^3)\). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.     

A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains

Fractional partial differential equations (PDEs) provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long-range spatial interactions, which cannot be modeled accurately by classical second-order diffusion equations. However, numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices, for which a direct solver requires $O\left(N^3\right)$ computations per time step and $O\left(N^2\right)$ memory, where $N$ is the number of unknowns. The significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three-dimensional space-fractional diffusion equations computationally intractable.Fast numerical methods were previously developed for space-fractional PDEs on multidimensional rectangular domains, without resorting to lossy compression, but rather, via the exploration of the tensor-product form of the Toeplitz-like decompositions of the stiffness matrices. In this paper we develop a fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions. The fast method has an optimal order storage requirement and almost linear computational complexity, without any lossy compression. Numerical experiments show the utility of the method.     

Mixed Methods for a Stream-Function – Vorticity Formulation of the Axisymmetric Brinkman Equations

This paper is devoted to the numerical analysis of a family of finite element approximations for the axisymmetric, meridian Brinkman equations written in terms of the stream-function and vorticity. A mixed formulation is introduced involving appropriate weighted Sobolev spaces, where well-posedness is derived by means of the Babu?ka–Brezzi theory. We introduce a suitable Galerkin discretization based on continuous piecewise polynomials of degree \(k\ge 1\) for all the unknowns, where its solvability is established using the same framework as the continuous problem. Optimal a priori error estimates are derived, which are robust with respect to the fluid viscosity, and valid also in the pure Darcy limit. A few numerical examples are presented to illustrate the convergence and performance of the proposed schemes.     

Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations

Optimal-order error estimates in the energy norm and the \(L^2\) norm were previously proved in the literature for finite element methods of Dirichlet boundary-value problems of steady-state fractional diffusion equations under the assumption that the true solutions have desired regularity and that the solution to the dual problem has full regularity for each right-hand side. We show that the solution to the homogeneous Dirichlet boundary-value problem of a one-dimensional steady-state fractional diffusion equation of constant coefficient and source term is not necessarily in the Sobolev space \(H^1\). This fact has the following implications: (i) Up to now, there are no verifiable conditions on the coefficients and source terms of fractional diffusion equations in the literature to ensure the high regularity of the true solutions, which are in turn needed to guarantee the high-order convergence rates of their numerical approximations. (ii) Any Nitsche-lifting based proof of optimal-order \(L^2\) error estimates of finite element methods in the literature is invalid. We present numerical results to show that high-order finite element methods for a steady-state fractional diffusion equation with smooth data and source term fail to achieve high-order convergence rates. We present a preliminary development of an indirect finite element method, which reduces the solution of fractional diffusion equations to that of second-order diffusion equations postprocessed by a fractional differentiation. We prove that the corresponding high-order methods achieve high-order convergence rates even though the true solutions are not smooth, provided that the coefficient and source term of the problem have desired regularities. Numerical experiments are presented to substantiate the theoretical estimates.     

Unconditional $$L^{\infty }$$-convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions

This paper is concerned with the unconditional and optimal \(L^{\infty }\)-error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrödinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a ‘lifting’ technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of \(O(h^4+\tau ^2)\) with time step \(\tau \) and mesh size h in the discrete \(L^{\infty }\)-norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrödinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.