NEW BOUNDARY ELEMENT FORMULATION FOR THE SOLUTION OF LAPLACE’S EQUATION
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The boundary element method (BEM) is typically used for solving potential problems and has several advantages over the traditional finite element method (FEM). However, the normal derivative of the potential appears explicitly as an unknown, with some inherent ambiguity at the corner nodes. Moreover, the BEM requires time consuming numerical integration (Gaussian quadrature) along the boundary of the domain. In this paper, we propose a new approach of BEM by introducing the weak nodal “cap” flux approach defined in the context of the finite element method. The domain integrals are eliminated at the discrete level by introducing the FEM approximation of the fundamental solutions at every node of the related mesh as basic functions in the Galerkin formulation of the BVP under study. The implementation of this new technique appears to be simpler as no numerical integration on the boundary of the domain is required so that the method leads to a substantially reduced computational burden. Our method is compared to the classical BEM for the numerical solution of the two-dimensional Laplace equation. It is observed that the normal flux presents a better behaviour at corners. A loss of accuracy may occur but it is compensated by a smaller execution time, allowing a finer mesh.
finite element method, boundary element method, green functions, electrostatics