Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations

In this paper we present a battery of results related to how Galerkin semidiscretization in space affects some formulations of wave scattering and propagation problems when retarded boundary integral equations are used.     

An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods

In this paper we analyze the coupling of local discontinuous Galerkin (LDG) and boundary element methods as applied to linear exterior boundary value problems in the plane. As a model problem we consider a Poisson equation in an annular polygonal domain coupled with a Laplace equation in the surrounding unbounded exterior region. The technique resembles the usual coupling of finite elements and boundary elements, but the corresponding analysis becomes quite different. In particular, in order to deal with the weak continuity of the traces at the interface boundary, we need to define a mortar-type auxiliary unknown representing an interior approximation of the normal derivative. We prove the stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Finally, we apply local and global approximation properties of the subspaces involved to obtain the a priori error estimate in the energy norm.

     

Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem

In this article we analyze the well‐posedness (unique solvability, stability, and Céa's estimate) of a family of Galerkin schemes for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers—Joseph—Saffman law. We consider the usual primal formulation in the Stokes domain and the dual‐mixed one in the Darcy region, which yields a compact perturbation of an invertible mapping as the resulting operator equation. We then apply a classical result on projection methods for Fredholm operators of index zero to show that use of any pair of stable Stokes and Darcy elements implies the well‐posedness of the corresponding Stokes—Darcy Galerkin scheme. This extends previous results showing well‐posedness only for Bernardi—Raugel and Raviart—Thomas elements. In addition, we show that under somewhat more demanding hypotheses, an alternative approach that makes no use of compactness arguments can also be applied. Finally, we provide several numerical results illustrating the good performance of the Galerkin method for different geometries of the problem using the MINI element and the Raviart—Thomas subspace of lowest order. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 721–748, 2011     

Periodic Dirac Delta Distributions in the Boundary Element Method

This paper is concerned with the numerical solution of boundary integral equations on smooth curves of the plane with some numerical methods having in common the use of sets of equally spaced periodic Dirac delta distributions as trial functions. In a functional frame of classical periodic pseudodifferential equations of nonpositive order, delta-spline and delta–delta methods are introduced and analysed with the overall aim of obtaining asymptotic expansions of the error in weak and strong norms. As a byproduct we obtain the convergence of the coefficients associated to the discrete delta approximation to pointwise values of the unknown, as well as superconvergent choices of positions of the delta distributions in relation with the discretization grid. Two numerical examples are explored to show nodal errors and the applicability of Richardson extrapolation.     

A non symmetric FVM-BEM coupling method

We consider the prototype model for flow and transport of a concentration in porous media in an interior domain and couple it with a diffusion process in the corresponding unbounded exterior domain. To guarantee mass conservation and stability with respect to dominating convection also for a discrete solution we introduce a non symmetric coupling of the vertex-centered finite volume method (FVM) and the boundary element method (BEM). BEM approximates the unbounded exterior problem which avoids truncation of the domain. One can also interpret that the (unbounded) exterior problem “replaces” the boundary conditions of the interior problem. We aim to provide a first rigorous analysis of the discrete system for different model parameters; existence and uniqueness, convergence, and a priori estimates. Numerical examples illustrate the strength of the chosen method which is computational cheaper than the previous three field FVM-BEM couplings. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dirac delta methods for Helmholtz transmission problems

In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms.      

Characterizing the inf-sup condition on product spaces

In this paper we establish characterization results for the continuous and discrete inf-sup conditions on product spaces. The inf-sup condition for each component of the bilinear form involved and suitable decompositions of the pivot space in terms of the associated null spaces are the key ingredients of our theorems. We illustrate the theory through its application to bilinear forms arising from the variational formulations of several boundary value problems. Dedicated to Professor Ivo Babuska on the occasion of his 82nd birthday. This research was partially supported by Centro de Modelamiento Matemático (CMM) of the Universidad de Chile, by Centro de Investigación en Ingenierí a Matemática (CI²MA) of the Universidad de Concepción, by FEDER/MCYT Project MTM2007-63204, and by Gobierno de Aragón (Grupo Consolidado PDIE).     

Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform

This paper is concerned with a heat diffusion problem in a half-space which is motivated by the detection of material defects using thermal measurements. This problem is solved by inverting the Laplace transform with respect to time on a contour in the complex plane using an exponentially convergent quadrature rule. This leads to a finite number of time-independent problems, which can be solved in parallel using boundary integral equation methods. We provide a full numerical analysis of this scheme on compact time intervals. Our results are formulated in a way that they can easily be used for other diffusion problems in exterior or interior domains.     

Local Expansions of Periodic Spline Interpolation with Some Applications

In this paper we prove some asymptotic expansions of the error of interpolation on equally spaced nodes with periodic smoothest splines of arbitrary degree on a uniform partition. We obtain a local expansion in terms of derivatives of the interpolate. Afterwards we apply this result to the asymptotic study of the numerical solution of periodic integral equations of the second kind by means of ϵ – collocation methods. We show some new superconvergence results and give particular forms of these expansions depending on the choices of the parameter ϵ. We finally give some numerical experiments, which corroborate the theory.