Papers by Matthias Maischak
Applied Numerical Mathematics, Aug 1, 2005
In this paper we analyze the hp-discretization of a boundary integral formulation for the Signori... more In this paper we analyze the hp-discretization of a boundary integral formulation for the Signorini contact problem of the Laplacian. We prove convergence of the bem Galerkin solution in the energy norm and obtain, under mild regularity assumptions, an a priori error estimate. Using a hierarchical subspace decomposition we derive a reliable and efficient a posteriori error estimate. Based on the hierarchical estimators we present a three-step hpadaptive algorithm and present numerical results which yield appropriate mesh refinements and polynomial degree distributions.
Springer eBooks, May 23, 2008
The hp-methods is a very efficient and accurate tool in modern computational mechanics. For a wid... more The hp-methods is a very efficient and accurate tool in modern computational mechanics. For a wide range of problems hp-techniques provide the exponential convergence rate of the discrete solution to the exact solution, while h-version and p-version give only an algebraic convergence rate. The finite element method is used commonly for numerical simulations of contact problems [3]. The boundary element techniques are relatively seldom used, despite some significant advantages [1]. In the boundary element method only the boundaries of the bodies are discretized, which reduces the dimension of the problem by one. This simplifies e.g. mesh generation significantly. Also, the number of unknowns in the problem is reduced greatly, but in contrast to finite elements, the Galerkin matrix will be dense due to nonlocal boundary integral operators.
Computer Methods in Applied Mechanics and Engineering, Feb 1, 2005
ABSTRACT We present residual based and p-hierarchical a posteriori error estimators for a Galerki... more ABSTRACT We present residual based and p-hierarchical a posteriori error estimators for a Galerkin method coupling finite elements and boundary elements for time–harmonic interface problems in electromagnetics; special emphasis is taken for the eddy current problem. The Galerkin discretization uses lowest order Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise linear functions on the interface boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in the terms of the error estimators as well. The estimators are derived from the defect equation using Helmholtz and Hodge decompositions. Numerical tests underline reliability and efficiency of the given error estimators yielding reasonable mesh refinements.
Journal of applied and numerical optimization, 2021
We analyze a finite element/boundary element procedure to solve a nonconvex contact problem for t... more We analyze a finite element/boundary element procedure to solve a nonconvex contact problem for the double-well potential. After relaxing the associated functional, the degenerate minimization problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which may then be solved numerically. The convergence of the Galerkin approximations to certain macroscopic quantities and a corresponding a posteriori estimate for the approximation error are discussed.
Numerische Mathematik, Oct 9, 2010
We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interfa... more We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L pand L 2-Sobolev spaces.
Computers & mathematics with applications, Oct 1, 2004
We analyze a least squares formulation for the numerical solution of second-order linear transmis... more We analyze a least squares formulation for the numerical solution of second-order linear transmission problems in two and three dimensions, which allow jumps on the interface. In a bounded domain the second-order partial differential equation is rewritten as a first-order system; the part of the transmission problem which corresponds to the unbounded exterior domain is reformulated by means of boundary integral equations on the interface. The least squares functional is given in terms of Sobolev norms of order-1 and of order 1/2. These norms are computed by approximating the corresponding inner products using multilevel preconditioners for a second-order elliptic problem in a bounded domain ~ and for the weakly singular integral operator of the single layer potential on its boundary 0~. As preconditioners we use both multigrid and BPX algorithms, and the preconditioned system has bounded or mildly growing condition number. Numerical experiments confirm our theoretical results. (~) 2004 Elsevier Ltd. All rights reserved.
Discrete Applied Mathematics, Feb 1, 2014
We present a quadratic programming problem arising from the p-version for a finite element method... more We present a quadratic programming problem arising from the p-version for a finite element method with an obstacle condition prescribed in Gauss-Lobatto points. We show convergence of the approximate solution to the exact solution in the energy norm. We show an a-priori error estimate and derive an a-posteriori error estimate based on bubble functions which is used in an adaptive p-version. Numerical examples show the superiority of the p-version compared with the h-version.
Applied Numerical Mathematics, Oct 1, 2006
We study a 2-level multiplicative Schwarz method for the p version Galerkin boundary element meth... more We study a 2-level multiplicative Schwarz method for the p version Galerkin boundary element method for a weakly singular integral equation of the first kind in 3D. We prove that the rate of convergence of the multiplicative Schwarz operator for the p version grows only logarithmically in p and is independent of h.
The hp-method on geometrical reflned meshes for 3D-BEM leads to exponential fast convergence on p... more The hp-method on geometrical reflned meshes for 3D-BEM leads to exponential fast convergence on polyhedral domains [4, 5] with respect to the number of unknowns. Using numerical quadrature exponential fast convergence can be still achieved for curved surfaces [2, 6], if the surface parametrisation is relatively cheap. Here we will make use of a general surface parametrisation, cf. [3], and we will allow non-matching grids, cf. [1], to simplify the connection of separately meshed parts of the surface. Emphasize is given to balancing the computational costs for evaluation of the surface parametrisation, incorporating the constraints on the spline space due to hanging nodes, with the exponential fast convergence of the method. We will present several examples where we obtain exponential fast convergence on complicated geometries.
Springer eBooks, Apr 29, 2007
ABSTRACT We analyze the h-p version of the BEM for Dirichlet and Neumann problems of the Lamé equ... more ABSTRACT We analyze the h-p version of the BEM for Dirichlet and Neumann problems of the Lamé equation on open surface pieces. With given regularity of the solution in countably normed spaces we show that the boundary element Galerkin solution of the h-p version converges exponentially fast on geometrically graded meshes. We describe in detail how to use an analytic integration for the computation of the entries of the Galerkin matrix. Numerical benchmarks correspond to our theoretical results.
International Journal for Numerical Methods in Engineering, Aug 9, 2011
In this paper we develop an a posteriori error analysis of a coupling of finite elements and boun... more In this paper we develop an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid-structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time-harmonic vibration. Our methods combine integral equations for the exterior fluid and finite element methods for the elastic structure. It is well-known that due to the reduction of the boundary value problem to boundary integral equations the solution is not unique in general. However, due to superposition of various potentials, we consider a boundary integral equation which is uniquely solvable and which avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures are considered; one is based on the non-symmetric formulation and the other one is based on a symmetric formulation. For both formulations we derive reliable residual a posteriori error estimates. From the estimators we compute local error indicators which allow us to develop an adaptive mesh refinement strategy. For the two dimensional case we perform an adaptive algorithm on triangles and for the three dimensional case we use hanging nodes on hexahedrons. Numerical experiments underline our theoretical results.
Applied Numerical Mathematics, 2013
We analyze the coupling of the dual-mixed finite element method with the boundary integral equati... more We analyze the coupling of the dual-mixed finite element method with the boundary integral equation method. The result is a new mixed scheme for the exterior Stokes problem. The approach is based on the introduction of both the flux and the strain tensor of the velocity as further unknowns, which yields a twofold saddle point problem as the resulting variational formulation. We show existence and uniqueness of the solution for the continuous and discrete formulations and provide the associated error analysis. In particular, the corresponding Galerkin scheme is defined by using piecewise constant functions and Raviart-Thomas spaces of lowest order. Most of our analysis makes use of an extension of the classical Babuška-Brezzi theory to a class of saddle-point problems. Also, we develop a-posteriori error estimates (of Bank-Weiser type) and propose a reliable adaptive algorithm to compute the finite elements solutions. Finally, several numerical results are given.
Applied Numerical Mathematics, Apr 1, 2012
This paper deals with a least-squares formulation of a second order transmission problem for line... more This paper deals with a least-squares formulation of a second order transmission problem for linear elasticity. The problem in the unbounded exterior domain is rewritten with boundary integral equations on the boundary of the inner domain. In the interior domain we treat a linear elastic material which can also be nearly incompressible. The least-squares functional is given in terms of theH −1 (Ω) and H 1/2 (Γ) norms. These norms are realized by solution operators of corresponding dual norm problems which are approximated using multilevel preconditioners.
Mathematical Methods in The Applied Sciences, Nov 25, 2008
We construct a novel hp‐mortar boundary element method for two‐body frictional contact problems f... more We construct a novel hp‐mortar boundary element method for two‐body frictional contact problems for nonmatched discretizations. The contact constraints are imposed in the weak sense on the discrete set of Gauss–Lobatto points using the hp‐mortar projection operator. The problem is reformulated as a variational inequality with the Steklov–Poincaré operator over a convex cone of admissible solutions. We prove an a priori error estimate for the corresponding Galerkin solution in the energy norm. Due to the nonconformity of our approach, the Galerkin error is decomposed into the approximation error and the consistency error. Finally, we show that the Galerkin solution converges to the exact solution as 𝒪((h/p)1/4) in the energy norm for quasiuniform discretizations under mild regularity assumptions. We solve the Galerkin problem with a Dirichlet‐to‐Neumann algorithm. The original two‐body formulation is rewritten as a one‐body contact subproblem with friction and a one‐body Neumann subproblem. Then the original two‐body frictional contact problem is solved with a fixed point iteration. Copyright © 2008 John Wiley & Sons, Ltd.
Computing, Jun 1, 1996
The hp-Version of the Boundary Element Method for Helmholtz Screen Problems. We study the boundar... more The hp-Version of the Boundary Element Method for Helmholtz Screen Problems. We study the boundary element method for weakly singular and hypersingular integral equations of the first kind on screens resulting from the Dirichlet and Neumann problems for the Helmholtz equation. It is shown that the hp-version with geometrical refined meshes converges exponentially fast in both cases. We underline our theoretical results by numerical experiments for the pure h-, p-versions, the graded mesh and the hp-version with geometrically refined mesh.
Applied Numerical Mathematics, Aug 1, 2010
In this paper an a-posteriori error estimate for non-linear coupled FEM-BEM equations is derived ... more In this paper an a-posteriori error estimate for non-linear coupled FEM-BEM equations is derived by using the Steklov-Poincaré operator and hierarchical basis techniques. We obtain "local" error indicators which are based on two-level subspace decompositions with the additive Schwarz operator. We present an algorithm for adaptive error control which is driven by these error indicators and numerical results are included.
Journal of Integral Equations and Applications, Feb 1, 2017
Time domain Galerkin boundary elements provide an efficient tool for numerical solution of bounda... more Time domain Galerkin boundary elements provide an efficient tool for numerical solution of boundary value problems for the homogeneous wave equation. We review recent advances in their a posteriori error analysis and the resulting adaptive mesh refinement procedures, as well as basic algorithmic aspects of these methods. Numerical results for adaptive mesh refinements are discussed in two and three dimensions, as are benchmark problems in a half-space related to the transient emission of traffic noise. 2010 AMS Mathematics subject classification. Primary 65N38, 65R20, 74J05. Keywords and phrases. Time domain boundary element method, error estimates, adaptive mesh refinements, sound radiation.
Numerical Linear Algebra With Applications, Sep 1, 1999
We propose and analyze efficient preconditioners for the minimum residual method to solve indefin... more We propose and analyze efficient preconditioners for the minimum residual method to solve indefinite, symmetric systems of equations arising from the h-p version of finite element and boundary element coupling. According to the structure of the Galerkin matrix we study two-and three-block preconditioners corresponding to Neumann and Dirichlet problems for the finite element discretization. In the case of exact inversion of the blocks we obtain bounded iteration numbers for the two-block Jacobi solver and O(h −3/4 p 3/2) iteration numbers for the three-block Jacobi solver. Here, h denotes the mesh size and p the polynomial degree. For the efficient two-block method we analyze the influence of various preconditioners which are based on further decomposing the trial functions into nodal, edge and interior functions. By further splitting the ansatz space with respect to basis functions associated with the edges we obtain a partially diagonal preconditioner. The penultimate method requires O(log 2 p) iterations whereas the latter method needs O(p log 2 p) iterations. Numerical results are presented which support the theory.
Numerische Mathematik, May 1, 2004
The hypersingular integral equation of the first kind equivalently describes screen and Neumann p... more The hypersingular integral equation of the first kind equivalently describes screen and Neumann problems on an open surface piece. The paper establishes a computable upper error bound for its Galerkin approximation and so motivates adaptive mesh refining algorithms. Numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh-refining. The numerical realisation requires the evaluation of the hypersingular integral operator at a source point; this and other details on the algorithm are included.
Letters in Mathematical Physics, Aug 1, 1992
All solutions to the consistency equations are determined which have to be satisfied by anomalies... more All solutions to the consistency equations are determined which have to be satisfied by anomalies in gravitational theories with a de Sitter-invariant groundstate. They turn out to be identical with the solutions for a Poincar6-invariant groundstate.
Uploads
Papers by Matthias Maischak