This chapter gives an expository introduction to the Galerkin-BEM for the elliptic boundary value problems from the mathematical point of view. Emphasis will be placed on the variational formulations of the boundary integral equations and the general error estimates for the approximate solution in appropriate Sobolev spaces. A classification of boundary integral equations will be given on the basis of the Sobolev index. The simple relations between the variational formulations of the boundary integral equations and the corresponding partial differential equations under consideration will be indicated. Basic concepts such as stability, consistency, and convergence, as well as the condition numbers and ill-posedness, will be discussed by using elementary examples.


Keywords:

boundary integral equations;
fundamental solutions;
variational formulations;
Garding's inequality;
Galerkin's method;
boundary elements;
stability;
ill-posedness;
asymptotic error estimates

Boundary Element Methods: Foundation and Error Analysis

Multigrid methods of the first kind are applied for solving iteratively some Symmetric algebraic Systems that oeeur in the numerical treatment of Symmetrie and strongly elliptic boundary integral eguations of nonnegative order. Using a damped Jacobi relaxation scheine, the iterative method proves to be unconditionally convergent in case of smooth boundaries and, moreover, convergent for a sufficiently large number of smoothing steps in case of general Lipschitz boundaries. As a numerical example, the Neumann problem of the Laplacean is treated by the “first kind” boundary element approach.

On Multigrid Methods of the First Kind for Symmetric Boundary Integral Equations of Nonnegative Order

This paper is concerned with the recent developments in the solution of boundary value problems by integral equations of the first kind. Basic results for weakly singular and hypersingular boundary integral operators will be discussed. Emphases will be given to the mathematical foundation of the method as well as to the physical interpretations of various side conditions derived for the unique solvability of the integral equations of the first kind.

Solution of Boundary Value Problems by Integral Equations of the First Kind — A Update

A discrete Galerkin boundary element technique with a quadratic approximation of the variables was developed to simulate the three-dimensional (3D) viscous flow established in periodic assemblages of particles in suspensions and within a periodic porous medium. The Batchelor's unit-cell approach is used. The Galerkin formulation effectively handles the discontinuity in the traction arising in flow boundaries with edges or corners, such as the unit cell in this case. For an ellipsoidal dilute suspension over the range of aspect ratio studied (1 to 54), the numerical solutions of the rotational velocity of the particles and the viscosity correction were found to agree with the analytic values within 0.2% and 2% respectively, even with coarse meshes. In a suspension of cylindrical particles the calculated period of rotation agreed with the experimental data. However, Burgers' predictions for the correction to the suspension viscosity were found to be 30% too low and therefore the concept of the equivalent ellipsoidal ratio is judged to be inadequate. For pressure-driven flow through a fixed bed of fibres, the prediction on the permeability was shown to deviate by as much as 10% from the value calculated based on approximate permeability additivity rules using the corresponding values for planar flow past a periodic array of parallel cylinders. These applications show the versatility of the technique for studying viscous flows in complicated 3D geometries.

Analysis of periodic 3D viscous flows using a quadratic discrete Galerkin boundary element method

Modelling of some elliptic fluid mechanics problems by the boundary element method

A larger class of two-dimensional elliptic boundary value problems in elasticity and fluid mechanics can be reduced to systems of boundary integral equations of the first kind. This paper is concerned with the stability analysis of boundary element methods for treating such a class of integral equations. In particular, the problem of ill-posedness, the optimal rate of convergence, and its connection with Tikhonov regularization procedure will be discussed.

On the Stability of Boundary Element Methods for Integral Equations of the First Kind

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On the Stability of Boundary Element Methods for Integral Equations of the First Kind