A weak formulation of boundary integral equations,with application to elasticity problems

A weak formulation for ‘direct’ boundary methods, deduced from distribution theory, is presented. The present approach seems particularly profitable when dealing with problems having non-integrable singularities. Numerical examples are also reported for plane elasticity.     

A non-local boundary value problem method for parabolic equations backward in time

The ill-posed parabolic equation backward in time

     

Global boundary value problems for nonlinear dynamic equations on time scales

We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.     

A PDAE formulation of parabolic problems with dynamic boundary conditions

The weak formulation of parabolic problems with dynamic boundary conditions is rewritten in form of a partial differential–algebraic equation. More precisely, we consider two dynamic equations with a coupling condition on the boundary. This constraint is included explicitly as an additional equation and incorporated with the help of a Lagrange multiplier. Well-posedness of the formulation is shown.     

The fast multipole method for the symmetric boundary integral formulation

^** Email: of{at}mathematik.uni-stuttgart.de^*** Email: o.steinbach{at}tugraz.at^**** Email: wendland{at}mathematik.uni-stuttgart.de A symmetric Galerkin boundary-element method is used for thesolution of boundary-value problems with mixed boundary conditionsof Dirichlet and Neumann type. As a model problem we considerthe Laplace equation. When an iterative scheme is employed forsolving the resulting linear system, the discrete boundary integraloperators are realized by the fast multipole method. While thesingle-layer potential can be implemented straightforwardlyas in the original algorithm for particle simulation, the double-layerpotential and its adjoint operator are approximated by the applicationof normal derivatives to the multipole series for the kernelof the single-layer potential. The Galerkin discretization ofthe hypersingular integral operator is reduced to the single-layerpotential via integration by parts. We finally present a correspondingstability and error analysis for these approximations by thefast multipole method of the boundary integral operators. Itis shown that the use of the fast multipole method does notharm the optimal asymptotic convergence. The resulting linearsystem is solved by a GMRES scheme which is preconditioned bythe use of hierarchical strategies as already employed in thefast multipole method. Our numerical examples are in agreementwith the theoretical results.     

A class of free boundary problems to quasi-linear parabolic equations

1.IntroductionStefan-likeproblemswithakineticconditiononthefreeboundarytolinearparabolicequationshavebeenconsideredbyseveralauthors(see[1-4]andreferencestherein),buttoquasi-linearonesproblemssimilar,whichareconsideredhereandmoredifficultthanpreviouso...     

Time-dependent boundary integral equations for multiply connected plates

The existence of distributional solutions is discussed for theinitial-boundary value problems associated with the motion ofa thin, elastic, multiply connected plate, and for the boundaryequations arising from integral representations of such solutions.     

Weak solutions for time‐dependent boundary integral equations associated with the bending of elastic plates under combined boundary data

The existence, uniqueness, stability, and integral representation of distributional solutions are investigated for the equations of motion of a thin elastic plate with a combination of displacement and moment‐stress components prescribed on the boundary. Copyright © 2004 John Wiley & Sons, Ltd.     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Spectral-fractional step Runge-Kutta discretizations for initial boundary value problems with time dependent boundary conditions

In this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting, alternating direction or, more generally, fractional step methods. Such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive Runge-Kutta method, which is called the fractional step Runge-Kutta method, and a standard space discretization which can be of type finite differences, finite elements or spectral methods among others. Spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps. The numerical experiences performed also show the increase of accuracy that this technique provides.

     

Some existence and uniqueness results for boundary value problems for difference equations

Existence and uniqueness results for bvp problems for difference equations are discussed. The weighted norm technique and the Banach contraction mapping principle are employed     

A stochastic approximation for fully nonlinear free boundary parabolic problems

We present a stochastic numerical method for solving fully nonlinear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the nonlinearity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 902–929, 2014     

Solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations at resonance

In this paper, we study the solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations with fractional derivative under resonant conditions. Firstly, the kernel function is presented through the Laplace transform and the properties of the kernel function are obtained. And then, some new results on the solvability for the boundary value problem are established by using Mawhin''s coincidence degree theory. Finally, two examples are presented to illustrate the applicability of our main results.     

A dual-dual mixed formulation for nonlinear exterior transmission problems

We combine a dual-mixed finite element method with a Dirichlet-to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

     

Spline collocation for convolutional parabolic boundary integral equations

Summary. We consider spline collocation methods for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover for example the case of the single layer heat operator equation when the spatial domain is a disc. Received December 15, 1997 / Revised version received November 16, 1998 / Published online September 24, 1999     

A boundary integral formulation of the Stefan problem

This paper considers a boundary integral formulation of the Stefan problem for two spatial dimensions. This formulation has the advantage that its numerical implementation does not require the discretization of the Stefan condition. Furthermore, the formulation is capable of solving problems with complex boundaries. Several illustrative examples are given.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.