An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR² via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.     

Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems

Summary The purpose of this article is to obtainL ² and uniform norm error estimates for the Galerkin approximation of the solution of certain boundary value problems via a comparison with then-norm projection of the solution. In some cases, these estimates constitute an improvement over known results.This research was supported in part by AEC Grant (11-1)-2075.     

On the energetic Galerkin boundary element method applied to interior wave propagation problems

We consider two-dimensional interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potential. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the related energetic Galerkin boundary element method. Numerical results are presented and discussed.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation error in the approximation of the nonlinear convective terms. The estimate of this error allows to analyse the error estimate of the method. The results obtained represent the completion and extension of the analysis from V. Dolej?í, M. Feistauer, Numer. Funct. Anal. Optim. 26 (2005), 349–383, where the truncation error in the approximation of the nonlinear convection terms was proved only in the case when the Dirichlet boundary condition on the whole boundary of the computational domain was considered.     

A domain embedding method for mixed boundary value problems

We propose a domain embedding (fictitious domain) method for elliptic equations subject to mixed boundary conditions, and prove the sharp convergence rate. The theory provides a unified treatment for Dirichlet, Neumann, and Robin boundary conditions. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.Dedicated to Prof. Dr.-Ing. W. L. Wendland on the occasion of his 50th birthday.A part of this work was done while the first author was a guest at the Georgia Institute of Technology and while the second author was partially supported by the NSF grant DMS-8501797.     

A direct discontinuous Galerkin finite element method for convection-dominated two-point boundary value problems

The direct discontinuous Galerkin (DDG) finite element method, using piecewise polynomials of degree k ≥ 1 on a Shishkin mesh, is applied to convection-dominated singularly perturbed two-point boundary value problems. Consistency, stability and convergence of order k (up to a logarithmic factor) are proved in an energy-type norm appropriate to the method and problem. The results are robust, i.e., they hold uniformly for all values of the singular perturbation parameter. Numerical experiments confirm the theoretical convergence rate.

     

A discrete Galerkin method for a hyperslngular boundary integral equation

Received on 14 August 1995. Revised on 20 August 1996. Consider solving the interior Neumann problem with D a simply-connected planar region and S=D a smooth curve.A double-layer potential is used to represent the solution,and it leads to the problem of solving a hypersingular integralequation. This integral equation is reformulated as a Cauchysingular integral equation. A discrete Galerkin method withtrigonometric polynomials is then given for its solution. Anerror analysis is given, and numerical examples complete thepaper.     

Second-order boundary value problems with integral boundary conditions

In this paper we investigate the existence of positive solutions of nonlocal second-order boundary value problems with integral boundary conditions.     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

A new method for solving singular fourth-order boundary value problems with mixed boundary conditions

In [1], [2], [3], [4], [5], [6], [7] and [8], it is very difficult to get reproducing kernel space of problem (1). This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth-order in the new reproducing kernel space. The numerical results are compared with both the exact solution and its n-order derived functions in the example. It is demonstrated that the new method is quite accurate and efficient for fourth-order problems.     

On the integral equation method for the plane mixed boundary value problem of the Laplacian

Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ₂ and given Neumann data on the remaining part Γ₂ of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ₂ and the also desired values of the normal derivatie on Γ₁. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ₁ ∩ Γ₂ even for C^∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ₂ providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ₁ and Γ₂ augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).     

On mixed boundary value problems for pseudoparabolic systems

Solvability analysis of mixed boundary value problems for pseudoparabolic systems in a special scale of weighted Sobolev spaces is presented. The class under consideration contains the linearized Navier-Stokes system. It is proved that, choosing the power weight, one can diminish the number of solvability conditions and in some cases obtain unconditional solvability of the boundary value problems.     

Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems

Numerical Algorithms - This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value...