A BDDC algorithm for the mortar-type rotated Q 1 FEM for elliptic problems with discontinuous coefficients

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

We propose a smooth fictitious domain/multiresolution method for enhancing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and uniqueness of the solution of a corresponding discrete problem and a so-called interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a Cassini oval.

On existence and uniqueness of solutions of boundary value problems of fourth order elliptic partial differential equations with variable coefficients

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

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, a stable extrinsic extended finite element method (SXFEM) is proposed to solve the second order elliptic equation with discontinuous coefficients and interfaces. SXFEM is designed by the stable enrichment function and stress intensity factors (SIF)-type enrichment functions. It shows that the proposed SXFEM can get the optimal convergence order. Numerical experiments are presented to verify the feasibility of the new method for this type of problem and the superiority compared with the standard FEM and XFEM.

In this paper, we investigate a two‐grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semilinear elliptic equation on the coarse mesh with mesh size , then, we use the coarse mesh solution as an initial guess to linearize the semilinear equation on the fine mesh, that is, on the fine mesh (with mesh size ), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution has sufficient regularity and , the two‐grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

We present an analysis of the additive average Schwarz preconditioner with two newly proposed adaptively enriched coarse spaces, which were presented at the twenty-third international conference on domain decomposition methods in Korea, for solving second-order elliptic problems with highly varying and discontinuous coefficients. It is shown that the condition number of the preconditioned system is bounded independently of the variations and the jumps in the coefficient while depending only on a prescribed threshold for the eigenvalues of the coarse space, and it depends linearly on the mesh parameter ratio H/h that is the ratio between the subdomain size and the mesh size thereby retaining the same optimality and scalability of the original additive average Schwarz preconditioner.

We compare a block successive overrelaxation method (BSOR) with a direct method for solving difference equations derived from fourth order elliptic partial differential equations. The direct method appears to be competitive in time and accuracy for a general problem.

Some multiscale methods for partial differential equations by J. Xu and A. Zhou Terascale implicit methods for partial differential equations by D. E. Keyes Boundary conditions for fully implicit two-phase flow models by M. Peszynska, E. W. Jenkins, and M. F. Wheeler Phase-field models of solidification by G. B. McFadden Singularity formation in free-surface Stokes flows by Q. Nie, S. Tanveer, T. F. Dupont, and X. Li A note on cache memory methods for multigrid in three dimensions by C. C. Douglas and D. T. Thorne.

We consider second order elliptic partial differential equations with coefficients that are singular or degenerate at an interior point of the domain. This paper presents formulation and analysis of a novel weighted-norm least squares finite element method for this class of problems. We propose a weighting scheme that eliminates the pollution effect and recovers optimal convergence rates. Theoretical results are carried out in appropriately weighted Sobolev spaces and include ellipticity bounds on the weighted homogeneous least squares functional, regularity bounds on the elliptic operator, and error estimates. Numerical experiments confirm the predicted error bounds.

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements.The Preconditioned Conjugate Gradient iterative method is used for solving of the large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. Both preconditioning techniques, displacement decomposition and circulant block-factorization, are highly parallelizable.A parallel algorithm is invented for the proposed preconditioner. The theoretical analysis of the execution time shows that the algorithm is highly efficient for coarse-grain parallel computer systems. A portable parallel FEM code based on MPI is developed. Numerical tests for real-life engineering problems in computational geomechanics are performed on a number of modern parallel computers: Cray T3E, Sunfire 6800, and Beowulf cluster. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.Keywordsparallel algorithmsPCG methodpreconditionercirculant matrixelasticity problem

Domain decomposition and expanded mixed method for parabolic partial differential equations

A classical problem in materials science involves the determination of effective transport properties in composites. The simplest problem is perhaps described by the second order elliptic partial differential equation where σ is piecewise constant, supplemented by suitable boundary conditions or by a radiation condition at infinity. This is a standard model for steady-state heat conduction and conductivity. For our purposes, it will be more convenient to rewrite (1) as an interface problem; we seek a potential function u which is harmonic in each component, continuous, and satisfies the continuity of flux condition