Exponential convergence of the hp-version for the boundary element method on open surfaces

Summary. We analyze the boundary element Galerkin method for weakly singular and hypersingular integral equations of the first kind on open surfaces. We show that the hp-version of the Galerkin method with geometrically refined meshes converges exponentially fast for both integral equations. The proof of this fast convergence is based on the special structure of the solutions of the integral equations which possess specific singularities at the corners and the edges of the surface. We show that these singularities can be efficiently approximated by piecewise tensor products of splines of different degrees on geometrically graded meshes. Numerical experiments supporting these results are presented. Received December 19, 1996 / Revised version received September 24, 1997 / Published online August 19, 1999     

Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes

Summary. We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional, order m on a bounded (open or closed) surface of dimension d, with . We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio. In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions. Received October 5, 2001 / Revised version received December 5, 2001 / Published online April 17, 2002 The support of this work through Visiting Fellowship grant GR/N21970 from the Engineering and Physical Sciences Research Council of Great Britain is gratefully acknowledged. The second author was also supported by the Australian Research Council     

Hybrid Galerkin boundary elements: theory and implementation

Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular) meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory. Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Wavelet approximations for first kind boundary integral equations on polygons

Summary. An elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation. Its Galerkin discretization with degrees of freedom on the boundary with spline wavelets as basis functions is analyzed. A truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from to entries. The condition numbers are bounded independently of the meshwidth. It is proved that the compressed scheme thus obtained yields in operations approximate solutions with the same asymptotic convergence rates as the full Galerkin scheme in the boundary energy norm as well as in interior points. Numerical examples show the asymptotic error analysis to be valid already for moderate values of . Received March 12, 1994 / Revised version received January 9, 1995     

A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems

We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes.     

On a hybrid boundary element method

Summary. In this paper we study a symmetric boundary element method based on a hybrid discretization of the Steklov–Poincaré operator well suited for a symmetric coupling of finite and boundary elements. The representation used involves only single and double layer potentials and does not require the discretization of the hypersingular integral operator as in the symmetric formulation. The stability of the hybrid Galerkin discretization is based on a BBL–like stability condition for the trial spaces. Numerical examples confirm the theoretical results. Received December 15, 1997 / Revised version received December 21, 1998/ Published online November 17, 1999     

Discrete boundary element methods on general meshes in 3D

Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on , for some . In contrast to other studies on this topic, we do not assume our meshes to be quasiuniform, and therefore the analysis admits locally refined meshes. To achieve such generality, standard inverse estimates for the quasiuniform case are replaced by appropriate generalised estimates which hold even in the locally refined case. Since the approximation of singular integrals on or near the diagonal of the Galerkin matrix has been well-analysed previously, this paper deals only with errors in the integration of the nearly singular and smooth Galerkin integrals which comprise the dominant part of the matrix. Our results show how accurate the quadrature rules must be in order that the resulting discrete Galerkin method enjoys the same stability properties and convergence rates as the true Galerkin method. Although this study considers only continuous piecewise linear basis functions on triangles, our approach is not restricted in principle to this case. As an example, the theory is applied here to conventional “triangle-based” quadrature rules which are commonly used in practice. A subsequent paper [14] introduces a new and much more efficient “node-based” approach and analyses it using the results of the present paper. Received December 10, 1997 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

Interior estimates for the wavelet Galerkin method

Summary. In this paper we derive an interior estimate for the Galerkin method with wavelet-type basis. Such an estimate follows from interior Galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems. We show that the error in an interior domain can be estimated with the best order of accuracy possible, provided the solution is sufficiently regular in a slightly larger domain, and that an estimate of the same order exists for the error in a weaker norm (measuring the effects from outside the domain ). Examples of the application of such an estimate are given for different problems. Received May 17, 1995 / Revised version received April 26, 1996     

Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h _min|)², where h _min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.     

Conditional ε-uniform boundedness of Galerkin projectors and convergence of an adaptive mesh method as applied to singularly perturbed boundary value problems

The Galerkin finite element method is applied to nonself-adjoint singularly perturbed boundary value problems on Shishkin meshes. The Galerkin projection method is used to obtain conditionally ε-uniform a priori error estimates and to prove the convergence of a sequence of meshes in the case of an unknown boundary layer edge.     

On the effect of numerical integration in the Galerkin boundary element method

Summary. We discuss the effect of cubature errors when using the Galerkin method for approximating the solution of Fredholm integral equations in three dimensions. The accuracy of the cubature method has to be chosen such that the error resulting from this further discretization does not increase the asymptotic discretization error. We will show that the asymptotic accuracy is not influenced provided that polynomials of a certain degree are integrated exactly by the cubature method. This is done by applying the Bramble-Hilbert Lemma to the boundary element method. Received May 24, 1995     

A new superconvergence property of Wilson nonconforming finite element

Summary. In this paper the Wilson nonconforming finite element method is considered to solve a class of two-dimensional second-order elliptic boundary value problems. A new superconvergence property at the vertices and the midpoints of four edges of rectangular meshes is obtained. Received May 5, 1995 / Revised version received November 11, 1996     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L _∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter _τ0 specifying the mesh. When _τ0 is too small, the error becomes O(1), but for _τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results. This revised version was published online in August 2006 with corrections to the Cover Date.     

Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement. Received August 1, 1996 / Revised version received February 28, 1997     

Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods Part II. The three-dimensional case

Summary. In this paper we introduce new local a-posteriori error indicators for the Galerkin discretization of three-dimensional boundary integral equations. These error indicators are efficient and reliable for a wide class of integral operators, in particular for operators of negative order. They are based on local norms of the computable residual and can be used for controlling the adaptive refinement. The proofs of efficiency and reliability are based on the result that the Aronszajn-Slobodeckij norm (given by a double integral for a non-integer ) is localizable for certain functions. Neither inverse estimates nor saturation properties are needed. In this paper, we extend the two-dimensional results of a previous paper to the three-dimensional case. Received March 20, 2000 / Published online November 15, 2001     

Additive Schwarz algorithms for the p version of the Galerkin boundary element method

Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators. Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000     

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.     

On the boundary element method for the Signorini problem of the Laplacian

Summary For the Laplace equation with Signorini boundary conditions two equivalent boundary variational inequality formulations are deduced. We investigate the discretization by a boundary element Galerkin method and obtain quasi-optimal asymptotic error estimates in the underlying Sobolev spaces. An algorithm based on the decomposition-coordination method is used to solve the discretized problems. Numerical examples confirm the predicted rate of convergence.