Self-adaptive finite-element techniques for the computation of inhomogeneous Poissonian fields

It is shown how complementary energy methods can be used both with the finite-element method to provide error bounds on solutions and as a basis for a self-adaptive mesh-generation strategy applicable to inhomogeneous Poissonian field problems. The self-adaptive approach allows very simple initial mesh structures to be defined, and yet the automatically generated mesh is such that optimal computer accuracy is guaranteed. This technique, suitable for solving electrostatic-field problems, is also highly computer efficient, particularly if the parameter of interest is system energy. Under such circumstances a close estimate to the exact solution can be obtained with the aid of two simple solutions using only a low number of degrees of freedom. This is a direct consequence of the error-bounded nature of the complementary forms. Results are presented which illustrate the operation of these techniques and their effectiveness