An elliptic singularly perturbed problem with two parameters II: Robust finite element solution

In this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square. The problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer-adapted Shishkin mesh. We prove that method with bilinear elements is uniformly convergent in an energy norm. Numerical results confirm our theoretical analysis.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

Asymptotische Fehlerschranken für Rayleigh-Ritz-Approximationen selbstadjungierter Eigenwertaufgaben

Zusammenfassung In der vorliegenden Arbeit leiten wir Fehlerschranken her für die Rayleigh-Ritz-Näherungen der Eigenwerte und Eigenelemente von selbstadjungierten Eigenwertaufgaben, deren Spektren nach unten beschränkt und anfangsdiskret sind. Solche Abschätzungen haben gerade im Hinblick auf ihre Bedeutung bei der Finite-Elemente-Methode eine lange Tradition, vgl. etwa [3-5, 7-9] sowie insbesondere die in [3] aufgeführte Literatur.Verfeinerte Fehlerschranken haben kürzlich Babuka und Osborn [1–3] angegeben: Speziell für mehrfach vorliegende Eigenwerte führen sie Approximationsgrößen ein, die die Güte der Diskretisierung beschreiben, und schätzen dann in Termen dieser Größen die Fehler ab. Dabei betrachten sie jedoch aus beweistechnischer Notwendigkeit heraus stets den Fall eines rein diskreten Spektrums, das sich durch selbstadjungierte und kompakte Operatoren beschreiben läßt. Wir lösen uns hier von dieser Einschränkung und geben in analoger Weise das asymptotische Verhalten der Fehler auch für solche Aufgaben an, die ein wesentliches Spektrum besitzen. Zugleich verbessern wir dabei die in [7] aufgestellten Schranken. Unsere Beweismethoden liefern explizit alle in den Fehlerschranken auftretenden Konstanten. Diese ergeben sich allein aus dem Spektrum der betrachteten Aufgabe. Desweiteren können wir alle Beweise mit reellwertigem Skalarkörper durchführen, da wir eine Darstellung von Spektralprojektoren durch Kurvenintegrale (vgl. etwa die Methoden der Spektralapproximation bei Chatelin [5]) für selbstadjungierte Probleme nicht benötigen. Die erhaltenen Ergebnisse gelten jedoch auch entsprechend für komplexwertige Skalarkörper.

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Asymptotic error estimates for Rayleigh-Ritz-approximations of selfadjoint eigenvalue problems

Summary In this paper we establish estimates for the approximation of the eigenvalues and eigenvectors of a selfadjoint eigenvalue problem bounded below with spectrum that begins with isolated eigenvalues of finite multiplicity. Results on the asymptotic behavior of the errors recently proved by Babuka and Osborn [1–3] are also valid for problems with nontrivial essential spectrum.

Herrn Professor Dr. Harro Heuser zum 65. Geburtstag gewidmet     

Zur Konvergenz des Rayleigh-Ritz-Verfahrens bei Eigenwertaufgaben

Zusammenfassung Bei der näherungsweisen Berechnung von Eigenwerten und Eigenelementen spielt das klassische Rayleigh-Ritz-Verfahren gerade im Hinblick auf die Finite-Elemente-Methode eine bedeutende Rolle. Umfangreiche Untersuchungen zur Konvergenz des Verfahrens liegen vor (vgl. etwa [1, 2, 4, 5, 7, 12, 15, 16, 19, 20]). Dabei werden zumeist quantitative Fehlerabschätzungen für konkrete Approximationen spezieller Probleme hergeleitet. Die Behandlung der Näherungseigenelemente erweist sich als besonders schwierig, wenn die zugehörigen Eigenwerte mehrfach vorliegen. Die für viele weiteren Untersuchungen fundamentalen Ergebnisse von Birkhoff et al. [1] gelten ausschließlich für einfache Eigenwerte.Das Hauptanliegen dieser Arbeit ist die Darlegung des qualitativen und quantitativen Konvergenzverhaltens der Näherungseigenelemente. In einer rein funktionalanalytischen Vorgehensweise betrachten wir Eigenwertaufgaben, die sich durch halbbeschränkte selbstadjungierte Operatoren beschreiben lassen. Die Konvergenzaussagen werden zurückgeführt auf die Approximationsgüte der Diskretisierung. Die erzielten Abschätzungen stehen somit generell fürjede Konkretisierung zur Verfügung. Insbesondere werden die Resultate aus [1] verallgemeinert. Die Beweise orientieren sich direkt an der Problemstellung. Methoden der Spektralapproximation (vgl. Chatelin [3], Vainikko [20]) werden nicht eingesetzt.

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On the convergence of the Rayleigh-Ritz method for eigenvalue problems

Summary This paper is concerned with the Rayleigh-Ritz method applied to eigenvalue problems, discribed by operators which are selfadjoint and bounded from below. In a purely functional analytic procedure the convergence results are reduced to error estimates for the discretization.

Dedicated to the memory of Professor Lothar Collatz     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks

The numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered. Since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods, based on global interpolation by analytic functions, show low accuracy. In order to circumvent this issue, a meshfree modification of the method of fundamental solutions is developed, where the approximation basis is enriched by an extra span of corner adapted non-smooth shape functions. The high accuracy of the new method is illustrated by solving several boundary value problems for the Helmholtz equation, modelling physical phenomena from the fields of room acoustics and acoustic resonance.     

Explicit expressions for three-dimensional boundary integrals in linear elasticity

On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lamé equation are included to validate the proposed formulae.     

On the super-approximation property of Galerkin's method with finite elements

Summary For Galerkin's method with finite elements as trial functions for strongly elliptic operator equations in the Hilbert scaleH ^t the super-approximation property and the optimal convergence rate are obtained by using the Aubin-Nitsche lemma. This applies in particular to spline collocation methods for a wide class of pseudodifferential equations.Dedicated to the memory of Professor Lothar Collatz     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

Solving Hankel matrix approximation problem using semidefinite programming

Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matrix to a given matrix to retrieve these properties. The problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second-order cone problem. The duality and optimality conditions are obtained and the primal–dual algorithm is outlined. Explicit expressions for a diagonal preconditioned and crossover criteria have been presented. Computational results are presented. A possibility for further improvement is indicated.     

A collocation method with cubic B-splines for solving the MRLW equation

The modified regularized long wave (MRLW) equation is solved numerically by collocation method using cubic B-splines finite element. A linear stability analysis of the scheme is shown to be marginally stable. Three invariants of motion are evaluated to determine the conservation properties of the algorithm, also the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.     

The convergence of spline collocation for strongly elliptic equations on curves

Summary Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.Dedicated to Prof. Dr. Dr. h.c. mult. Lothar Collatz on the occasion of his 75th birthdayThis work was begun at the Technische Hochschule Darmstadt where Professor Arnold was supported by a North Atlantic Treaty Organization Postdoctoral Fellowship. The work of Professor Arnold is supported by NSF grant BMS-8313247. The work of Professor Wendland was supported by the Stiftung Volkswagenwerk     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL ²,H ¹ andL , provided that certain relationships hold between the frequency, mesh size and outer radius.     

Discrete maximum principle for FE solutions of the diffusion-reaction problem on prismatic meshes

In this paper we analyse the discrete maximum principle (DMP) for a stationary diffusion-reaction problem solved by means of prismatic finite elements. We derive geometric conditions on the shape parameters of the prismatic partitions which guarantee validity of the DMP. The presented numerical tests show the sharpness of the obtained conditions.     

Numerical schemes for a pseudo-parabolic Burgers equation: Discontinuous data and long-time behaviour

We consider a simplified model for vertical non-stationary groundwater flow, which includes dynamic capillary pressure effects. Specifically, we consider a viscous Burgers-type equation that is extended with a third-order term containing mixed derivatives in space and time. We analyse the one-dimensional boundary value problem and investigate numerically its long-time behaviour. The numerical schemes discussed here take into account possible discontinuities of the solution.     

An iterative adaptive finite element method for elliptic eigenvalue problems

We consider the task of resolving accurately the n$n$th eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n$n$ eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue–eigenfunction pair on a sequence of locally refined meshes. Both Picard’s and Newton’s variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp$hp$-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided.     

Boundary-variation solution of eigenvalue problems for elliptic operators

We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated far outside their radii of convergence—by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.     

Eigenfrequencies of nonisotropic fractal drums

Let Γ be a regular nonisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalue of the fractal differential operator

(-Δ)^-1tr^Γ