Adaptive Coupling and Fast Solution of FEM–BEM Equations for Parabolic–Elliptic Interface Problems

In this paper we prove an a posteriori error estimate for the symmetric coupling of finite elements and boundary elements applied to linear parabolic–elliptic interface problems. The discontinuous Galerkin method is used for the discretization in time. We present an adaptive algorithm for choosing the mesh size in space and time and we analyse the Hybrid Modified Conjugate Residual (HMCR) method as a solution method for the linear systems which arise. Computational results show that the number of HMCR-iterations grows slowly with the problem size. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Some Nonlinear Elliptic Problems in Unbounded Domains

In this paper the results of some investigations concerning nonlinear elliptic problems in unbounded domains are summarized and the main difficulties and ideas related to these researches are described. The model problem

Interface Problems for Elliptic Systems

Interface problems for elliptic systems of second order partial differential equations are studied. The main result is that the solution in the neighborhood of the singular point can be divided into two parts one of which is a solution to the homogeneous system with constant coefficients, and the other one possesses higher regularity.     

Discrete Fourier–Neumann series

Let J_μ denote the Bessel function of order μ. The systemwith n=0,1,…,α>−1, and where p_s denotes the sth positive zero of J_α(ax), is orthonormal in . In this paper, we study the mean convergence of the Fourier series with respect to this system. Also, we describe the space in which the span of the system is dense.     

Elliptic Obstacle Problems with Measurable Coefficients in Non-Smooth Domains

We establish a global weighted W ^{1, p} -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains.     

Uniform Convergence of Fourier–Jacobi Series

Necessary and sufficient conditions which imply the uniform convergence of the Fourier–Jacobi series of a continuous function are obtained under an assumption that the Fourier–Jacobi series is convergent at the end points of the segment of orthogonality [−1,1]. The conditions are in terms of the modulus of continuity, Λ-variation, and the modulus of variation of a function.     

Two Dimensional Interface Problems for Elliptic Equations

We study the structure of solutions to the interface problems for second order quasi-linear elliptic partial differential equations in two dimensional space. We prove that each weak solution can be decomposed into two parts near singular points, a finite sum of functions in the form of cr^α log^m rφ(θ) and a regular one w. The coefficients c and the C^{1,α} norm of w depend on the H¹-norm and the C^{º,α}-norm of the solution, and the equation only.     

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹( _s ) [W _m ¹( _s )]^* in a sequence of perforated domains _s . Under a certain condition imposed on the local capacity of the set \ _s , we prove the following principle of compensated compactness: , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹() and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from whose weak limit is equal to zero.     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.     

Multi-Bubble Solutions for Slightly Super-Critical Elliptic Problems in Domains with Symmetries

The aim of this paper is to show the existence of solutionswith an arbitrarily large number of bubbles for the slightlysuper-critical elliptic problem in , subject to the conditions that u > 0 in , and u = 0on , where > 0 is a small parameter and R^N is a boundeddomain with certain symmetries, for instance an annulus or atorus in R³. 2000 Mathematics Subject Classification 35J25 (primary);35J20, 35J60 (secondary).     

Mixed Boundary Value Problems for Nonlinear Elliptic Systems with p‐Structure in Polyhedral Domains

The nonlinear elliptic system is investigated on a non‐smooth domain. Mixed boundary value conditions are given. The left‐hand side of the system has p‐structure (e.g., it is the p‐Laplacian and 1 < p < ∞). Global regularity results of u and |∇u|^p/2 in fractional order Sobolev spaces are proven.     

一类无界区域中的椭圆型系统非局部边值问题

本文讨论了一类在无界区域中的非线性椭圆系统的非局部边值问题。在适当的条件下,相对边值问题解的存在性及其比较定理作了研究。     

An Error Estimate for a Finite-element Approximation of an Elliptic Variational Inequality Formulation of a Hele--Shaw Moving-boundary Problem

It is the purpose of this paper to estimate the error in thefinite-element approximation of a variational inequality formulationof a moving-boundary problem arising in Hele-Shaw flow. Linearfinite elements on a triangulation of a polygonal approximationof the domain are used. The novelty of the problem lies in thetime dependence, the semi-coercivity of the bilinear form andthe change in domain. To analyse the approximation, operatorsare introduced which split the space H1. Order h convergenceis proved in the H¹-Sobolev norm.     

On the Existence of Positive Solutions for Some Semilinear Elliptic Problems in Annular Domains

Results in Mathematics -     

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established and some numerical experiments, which confirm the theoretical results, are performed.The first two authors were supported by Ministerio de Ciencia y Tecnología (Spain). The second author was also supported by the DFG research center “Mathematics for key technologies” (FZT86) in Berlin.