Non-Conforming Domain Decomposition with Hybrid Method

We present a non-conforming domain decomposition technique for solving elliptic problems with the finite element method. Functions in the finite element space associated with this method may be discontinuous on the boundary of subdomains. The sizes of the finite meshes, the kinds of elements and the kinds of interpolation functions may be different in different subdomains. So, this method is more convenient and more efficient than the conforming domain decomposition method. We prove that the solution obtained by this method has the same convergence rate as by the conforming method, and both the condition number and the order of the capacitance matrix are much lower than those in the conforming case.     

杂交有限元的区域分解法

近几年来,由于并行计算机的迅速发展,求解椭圆型微分方程的区城分解法又引起了人们的重视.这一方法的基本思想是把求解区域分解成许多子区域,每个子区域上用一台计算机求解.这种方法适应并行机的需要,是用并行机解大型椭圆型偏微分方程的一     

小周期型复合材料稳态热传导问题的一种双尺度渐近展开收敛性分析

给出一类具有某种对称性小周期复合材料稳态热传导问题解的渐近表示方法.区别于传统多尺度计算方法,将计算过程中需要求解的关于单胞Q的H_per¹(Q)周期边值问题改为齐次边值问题,这样数值方法求解时协调元空间容易构造;另一方面传统的多尺度渐近解不满足原始问题的边界条件,新构造的渐近形式不仅满足原始问题的物理边界条件,同时保持一定的收敛阶,更能被工程上所接受.     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

An immersed finite element space and its approximation capability

This article discusses an immersed finite element (IFE) space introduced for solving a second‐order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body‐fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004     

Quadrature for meshless Nitsche's method

In this article, we study effect of numerical integration on Galerkin meshless method (GMM), applied to approximate solutions of elliptic partial differential equations with essential boundary conditions (EBC). It is well‐known that it is difficult to impose the EBC on the standard approximation space used in GMM. We have used the Nitsche's approach, which was introduced in context of finite element method, to impose the EBC. We refer to this approach as the meshless Nitsche's method (MNM). We require that the numerical integration rule satisfies (a) a “discrete Green's identity” on polynomial spaces, and (b) a “conforming condition” involving the additional integration terms introduced by the Nitsche's approach. Based on such numerical integration rules, we have obtained a convergence result for MNM with numerical integration, where the shape functions reproduce polynomials of degree k ≥ 1. Though we have presented the analysis for the nonsymmetric MNM, the analysis could be extended to the symmetric MNM similarly. Numerical results have been presented to illuminate the theoretical results and to demonstrate the efficiency of the algorithms.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 265–288, 2014     

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015     

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM

Averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing. In the second paper of our analysis of their reliability, we consider conforming -FEM of higher (i.e., not of lowest) order in two or three space dimensions. In this paper, reliablility is shown for conforming higher order finite element methods in a model situation, the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of local averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

     

Reduced Finite Element Discretizations of the Stokes and Navier-Stokes Equations

ABSTRACT

If finite element spaces for the velocity and pressure do not satisfy the Babu?ka-Brezzi condition, a stable conforming discretization of the Stokes or Navier-Stokes equations can be obtained by enriching the velocity space by suitable functions. Writing any function from the enriched space as a sum of a function from the original space and a function from the supplementary space, the discretization will contain a number of additional terms compared with a conforming discretization for the original pair of spaces. We show that not all these terms are necessary for the solvability of the discrete problem and for optimal convergence properties of the discrete solutions, which is useful for saving computer memory and for establishing a connection to stabilized methods.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.     

A robust preconditioner for higher order finite element discretizations in linear elasticity

Based on the auxiliary space method, a preconditioner is studied in this paper for linear systems of equations arising from higher order finite element (FEM) discretizations of linear elasticity equations. The main idea, which is proposed by Xu (Computing 1996; 56 :215–235) for the scalar PDE, is to construct the preconditioner as a combination of a smoother and a coarse level solver, where the systems of equations arising from lower order FEM discretizations are used in the coarse level solver. It is theoretically shown that the condition number of the preconditioned systems is uniformly bounded with respect to both the problem size and moderate Poisson's ratio. When the Poisson's ratio is near the limit of 0.5, we have presented some numerical tests for the case of fourth‐order FEM discretization in a combination with quadratic conforming FEM as a coarse space. The results are almost robust when Poisson's ratio is near the limit of 0.5. Copyright © 2011 John Wiley & Sons, Ltd.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

Domaines fictifs, éléments finis H(div) et condition de Neumann: le problème de la condition inf-sup

Approximation of the Neumann problem for a second order elliptic operator by a fictitious domain method with a Lagrange multiplier on the boundary is considered. The problem is written in its vectorial dual formulation and H(div) mixed finite elements for the vector unknown and H^1/2 conforming elements for the multiplier are used. The uniform inf-sup condition is demonstrated under a compatibility condition between surface and volume meshes.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

In this paper, an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods. In order to efficiently solve the generated linear large-scale system, the generalized minimal residual (GMRES) algorithm is applied. For accelerating the convergence rate of the iterative, the Strang-type, Chan-type and P-type preconditioners are introduced. The suggested method can reach higher order accuracy both in space and in time than the existing methods. When the used boundary value method is $A_{k1,k2}$-stable, it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1. It implies that the iterative solution is convergent rapidly. Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.     

Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible multiscale decompositions for both the domain and its boundary, and on the possibility of characterizing various function spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis. An explicit construction of the wavelet bases and the lifting is proposed on fairly general domains, based on conforming domain decomposition techniques. Received November 2, 1998 / Published online April 20, 2000     

Inf‐sup stability of Petrov‐Galerkin immersed finite element methods for one‐dimensional elliptic interface problems

In this article, we analyze the Petrov‐Galerkin immersed finite element method (PG‐IFEM) when applied to one‐dimensional elliptic interface problems. In the PG‐IFEM (T. Hou, X. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185‐205, and S. Hou and X. Liu, J. Comput. Phys., 202 (2005), 411‐445), the classic immersed finite element (IFE) space was taken as the trial space while the conforming linear finite element space was taken as the test space. We first prove the inf‐sup condition of the PG‐IFEM and then show the optimal error estimate in the energy norm. We also show the optimal estimate of the condition number of the stiffness matrix. The results are extended to two dimensional problems in a special case.     

Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions

We extend the applicability of the augmented dual-mixed method introduced recently in Gatica (2007), Gatica et al. (2009) to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neumann boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.     

The factorization method for cracks in inhomogeneous media

We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.