The stable extrinsic extended finite element method for second order elliptic equation with interfaces

In this paper, a modified intrinsic extended finite element method (XFEM) for one-dimensional and two-dimensional elliptic equations with discontinuous coefficients and interfaces is proposed. We improve the intrinsic XFEM by changing the shape functions of the critical nodes. The improved shape functions can be used to catch the discontinuous information near interfaces. In addition, we modify the Gauss integration in special elements cut by interfaces. Numerical experiments are presented to verify the feasibility and superiority of the modified intrinsic XFEM compared with the standard FEM and extrinsic XFEM for this type of problem. Results also show that the modified intrinsic XFEM can generate an approximate solution whose error is $$O(h^2)$$ in an $$\hbox {L}^2$$ -norm and O(h) in an energy norm if the Q1 element is used.

In this paper, we propose a BDDC preconditioner for the mortar-type rotated finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis. MSC:65N55, 65N30.

Analysis of the computational potential is provided for a modification of one of the numerical approaches to the classical boundary integral equations method. The originally proposed name of the approach is the method of functional equations, but in nowadays it is also referred to as the fundamental solutions method. Undesired after effects of this name flip are pointed out. The modification, implemented in this study, requires computer-friendly representations for some relevant Green's function and significantly enhances the resolving potential of the method. This work focuses on the exploration of the computational applicability of this modification. Chosen for that boundaryvalue problems and stated on regions of irregular configuration for second order elliptic equations with discontinuous coefficients.

A fast unconditionally stable time domain finite element (FUS-FETD) method based on local eigenvalue solutions is proposed in this paper. This method can reach unconditional stability for the explicit time marching, but it does not need to solve partial solutions of global eigenvalue problem. By strictly grouping and numbering, the unknowns in the computational domain are divided into two groups: the unknowns with fine grids and the unknowns without fine grids. The system matrix can be naturally divided into four different matrix blocks. By solving the eigenvalue problem of local matrix block related to fine grids, the unstable modes can be quickly and effectively obtained. Since the dimension of local matrix block is smaller than that of global matrix, the proposed method reduces the complexity of solving eigenvalue problem and improves the computational efficiency. Numerical results show the effectiveness of the proposed algorithm.

Parallel finite difference schemes for heat equation based upon overlapping domain decomposition

We study the second order elliptic equation with discontinuous intermediate coefficients. The variable coefficients and the right-hand side of this equation belong to some space M of type F. We find necessary and sufficient conditions on M for continuous differentiability of the solution to the above equation, when M is a space of type F or a symmetric space, or one of a Sobolev or a Besov space.

In particular our chief concern will be with the differentiability properties of generalized solutions of (1.1) in the neighborhood of a point on an arc across which the coefficients of (1.1) have finite jump discontinuities. A well-known local result [1], [5] may be stated as follows: If the coefficients of (1.1) are Holder continuous in G, the closure of a disk G with center at a point to, then there exists a closed disk G1 with center at to and G1CG such that the first derivatives of any generalized solution(2) w(z) of (1.1) in G are H6lder continuous in U1. Suppose now that G is separated into two parts G+ and Gby a sufficiently simple arc L passing through to, and that the coefficients A(z), B(z), C(z), and D(z) are Holder continuous in each of the closed regions G+ and Gand therefore have at most finite jump discontinuities across L. In Theorem 1, it is shown that there exists a suitably chosen disk Go with center at to and GoCG, such that the first derivatives w, and wd of any generalized solution of (1.1) are Holder continuous in each of the two closed regions Go+ and Go separately. This says that the derivatives w, and w2 are smooth up to L in Go from either side of L and have at most finite jump discontinuities across it. In ?4 we specialize (1.1) to Beltrami's equation

Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient

In this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for both the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. The numerical examples cover a wide range of applied problems, including singularly...

The second order elliptic equation, which is also know as the diffusion-convection equation, is of great interest in many branches of physics and industry. In this paper, we use the weak Galerkin finite element method to study the general second order elliptic equation. A weak Galerkin finite element method is proposed and analyzed. This scheme features piecewise polynomials of degree $k\geq 1$ on each element and piecewise polynomials of degree $k-1\geq 0$ on each edge or face of the element. Error estimates of optimal order of convergence rate are established in both discrete $H^1$ and standard $L^2$ norm. The paper also presents some numerical experiments to verify the efficiency of the method.

Unconditionally stable mixed finite element methods for Darcy flow

Galerkin methods for the Davey–Stewartson equations

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods for u′(t) = au(t) + bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.

This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular H(div) system. Numerical experiments also justify the robustness and efficiency of this scheme.

SummarySingularly perturbed second order elliptic equations with boundary layers are considered. Numerical methods composed of central difference operators on special piece-wise uniform meshes are constructed for the above problems. Numerical results are obtained which show that these methods give approximate solutions with error estimates that are independent of the singular perturbation parameter.