Will the boundary(-domain) integral-equation methods survive? Preface to the special issue on non-traditional boundary (-domain) integral-equation methods

Abstract

The Boundary Integral Equation (BIE) method (boundary element method, elastic potential method) has been intensively developed over recent decades both in theory and in engineering applications. Its popularity was due to reducing a Boundary Value Problem (BVP) for a partial differential equation in a domain to an integral equation on the domain boundary, that is, to diminishing the problem dimensionality by one. The main ingredient necessary for the reduction of a BVP to a BIE is a fundamental solution to the original partial differential equation. Employing the fundamental solution in the corresponding Green formula, one can reduce the problem to a boundary integral equation. After an appropriate discretization, this leads to a relatively small system of linear algebraic equations, which can be solved using small computer resources. In spite of these evident advantages, the popularity of BIE method does not look high nowadays. Although BIEs have their established niche in problems for infinite or semi-infinite domains with constant coefficients, appearing, e.g., in geomechanics, acoustics, fluid mechanics and some other engineering applications, the computational mechanics market is dominated by the Finite Element Method (FEM), at least in solid mechanics. Several reasons for this are listed below. First, the matrix of the linear algebraic equation system obtained after BIE discretization, is dense, while for FEM it is sparse and moreover, the number of non-zero entries in each of the FEM equations is determined by the element