On the sufficient and necessary BIEs

Abstract

In this talk, we examine the sufficient and necessary BIE for the unique solution of boundary value problems. The equivalence of the solution space for the BIE and the PDE as well as the direct and indirect BIEs will be examined. Revisit of the Fredholm alternative theorem is done by using the SVD technique. Although nonuniqueness solutions in the BEM/BIEM may appear in the degenerate boundary, degenerate scale, spurious eigenvalue and fictitious frequency, we only focus on the degenerate scale. Three detecting indexes of degenerate scale, the minimum singular value of the influence matrix, the logarithmic capacity in complex variables and free constant in the Fichera's approach, are proposed. Then, five regularization techniques, hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium, CHEEF method and Fichera's approach are addressed. Two analytical examples (circle and ellipse) were analytically studied by using the degenerate kernels. One numerical example of general shape was illustrated to see the validity of the proposed formulation.