Trefftz indirect method applied to nonlinear potential problems
Boundary type solution methodologies for linear elliptic boundary value problems (BVPs) are now well established. Unfortunately, this class of problems represents very few realistic practical situations due to the presence of nonlinearities in either boundary conditions or material properties. In this paper, we demonstrate an indirect formulation of the Trefftz method for solution of nonlinear BVPs arising in potential problems. While the concept of Trefftz method itself is fairly established, its application to nonlinear BVPs has been quite limited. Past experience indicates that Trefftz complete functions often give rise to polynomial type of nonlinearities and the resulting equations that govern the unknowns are almost always poorly conditioned, for which numerical solutions are difficult to obtain from standard procedures. For the solution of such nonlinear equations, we adopt a highly robust stabilized continuation method with inherent numerical error attenuation properties. Our numerical examples demonstrate this methodology when applied to 2D heat conduction problems with material and boundary condition nonlinearities, and the results indicate good potential for the use of indirect Trefftz method for nonlinear problems.
trefftz indirect method nonlinear potential problems boundary value problems stabilized continuation