Boundary Integral Solution of Quasi-linear Laplace Equation

For non-homogeneous or nonlinear problems, a major difficulty in applying the Boundary Element Method is the treatment of the volume integrals that arise. A recent proposed method, the grid-based integration method (GIM), uses a 3D uniform grid to efficiently perform volume integration. The efficiency of the GIM has been demonstrated on 3D Poisson problems. In this paper, we report our work on the extension of this technique to quasilinear problems. Numerical results of a 3D Heimholtz problem and a quasilinear Laplace problem on a solid sphere domain and a multiply-connected domain with Dirichlet boundary conditions are compared with analytic solutions. The performance of the GIM is measured by plotting the L2-norm error as a function of the overall CPU time and is compared with the auxiliary domain method in the Heimholtz problem.