A coupled BEM and FEM for the interior transmission problem in acoustics

The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method.     

Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains

In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms.     

An adaptive displacement/pressure finite element scheme for treating incompressibility effects in elasto-plastic materials

In this article, a mixed finite element formulation is described for coping with (nearly) incompressible behavior in elasto-plastic problems. In addition to the displacements, an auxiliary variable, playing the role of a pressure, is introduced resulting in Stokes-like problems. The discretization is done by a stabilized conforming Q1/Q1-element, and the corresponding algebraic systems are solved by an adaptive multigrid scheme using a smoother of block Gauss–Seidel type. The adaptive algorithm is based on the general concept of using duality arguments to obtain weighted a posteriori error bounds. This procedure is carried out here for the described discretization of elasto-plastic problems. Efficiency and reliability of the proposed adaptive method is demonstrated at (plane strain) model problems. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:369–382, 2001     

Benchmark results for testing adaptive finite element eigenvalue procedures

A discontinuous Galerkin method, with hp-adaptivity based on the approximate solution of appropriate dual problems, is employed for highly-accurate eigenvalue computations on a collection of benchmark examples. After demonstrating the effectivity of our computed error estimates on a few well-studied examples, we present results for several examples in which the coefficients of the partial-differential operators are discontinuous. The problems considered here are put forward as benchmarks upon which other adaptive methods for computing eigenvalues may be tested, with results compared to our own.     

Solving parabolic problems with different time steps in different regions in space based on domain decomposition methods

We describe a method for solving parabolic partial differential equations (PDEs) using local refinement in time. Different time steps are used in different spatial regions based on a domain decomposition finite element method. Extrapolation methods based on either a linearly implicit mid-point rule or a linearly implicit Euler method are used to integrate in time. Extrapolation methods are a better fit than BDF methods in our context since local time stepping in different spatial regions precludes history information. Some linear and nonlinear examples demonstrate the effectiveness of the method.     

A new model for the analysis of reinforced concrete members with a coupled HdBNM/FEM

As a truly boundary-type meshless method, the hybrid boundary node method (HdBNM) does not require ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. In this paper, the HdBNM is coupled with the finite element method (FEM) for predicting the mechanical behaviors of reinforced concrete. The steel bars are considered as body forces in the concrete. A bond model is presented to simulate the bond-slip between the concrete and steels using fictitious spring elements. The computational scale and cost for meshing can be further reduced. Numerical examples, in 2D and 3D cases, demonstrate the efficiency of the proposed approach.     

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 posteriori boundary control for FEM approximation of elliptic eigenvalue problems

We derive new a posteriori error estimates for the finite element solution of an elliptic eigenvalue problem, which take into account also the effects of the polygonal approximation of the domain. This suggests local error indicators that can be used to drive a procedure handling the mesh refinement together with the approximation of the domain. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 369–388, 2012     

Finite element approximation with numerical integration for differential eigenvalue problems

Error estimates of the finite element method with numerical integration for differential eigenvalue problems are presented. More specifically, refined results on the eigenvalue dependence for the eigenvalue and eigenfunction errors are proved. The theoretical results are illustrated by numerical experiments for a model problem.     

Solution of non-linear thermal transient problems by a new adaptive time-step method in quenching process

Quenching process is a thermo-elastic-plasticity problem with a high material non-linearity. The numerical oscillation is likely caused in the simulation of quenching process. In order to avoid the numerical oscillation and improve the calculation accuracy of temperature and phase-transformation fields in the quenching process, a new self-adaptive time-step size method is presented. The method can adjust the time-step size according to the maximum and minimum differences of temperature fields between the previous simulation step and the current simulation step. FEM software for evaluating the temperature, stress/strain and phase-transformation is also developed. A cooling example with numerical analytical results and a quenching example with experiment results are used to verify the calculation accuracy of this software. Five methods including the method in this paper, two constant time-step sizes and two geometric proportion time-step sizes are applied to simulate the quenching process of a 40Cr steel cylinder, respectively. A comparison of the simulation results shows that, the method presented in this paper can effectively avoid the numerical oscillation, ensure the calculation accuracy and cost less calculation time.     

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

定常Navier—Stokes方程的Petrov—Galerkin方法

研究了定常Navier－Stokes方程的四种Petrov－Galerkin有限元方法：PG1，PG2，SD和GLS．它们都是稳定的，避免了经典混合方法中必要的Babuska－Brezzi条件．给出了各种方法有限元解的存在性、唯一性和唯一解的误差估计．     

A finite element method for interface problems in domains with smooth boundaries and interfaces

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.     

An error indicator for mortar element solutions to the Stokes problem

We are interested in the mortar spectral element discretizationof the Stokes problem in a two-dimensional polygonal domain.We propose a family of residual type error indicators and weprove estimates which allow us to compare them with the error.We explain how these indicators can be used for adaptivity,and we present some numerical experiments.     

Analysis of a FEM for a coupled system of singularly perturbed reaction–diffusion equations

We consider a system of ℓ ≥ 2 one-dimensional singularly perturbed reaction–diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We present a convergence theory for conforming linear finite elements on arbitrary meshes. As a result convergence independently of the perturbation parameters on a wide class of layer-adapted meshes is established.      

Convergence analysis of an upwind finite volume element method with Crouzeix-Raviart element for non-selfadjoint and indefinite problems

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H ¹-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption      

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.     

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.     

On the finite element method for elliptic problems with degenerate and singular coefficients

We consider Dirichlet boundary value problems for second order elliptic equations over polygonal domains. The coefficients of the equations under consideration degenerate at an inner point of the domain, or behave singularly in the neighborhood of that point. This behavior may cause singularities in the solution. The solvability of the problems is proved in weighted Sobolev spaces, and their approximation by finite elements is studied. This study includes regularity results, graded meshes, and inverse estimates. Applications of the theory to some problems appearing in quantum mechanics are given. Numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi-uniform meshes.