A $p$ -Adaptive Scheme for Scalar Fields, Using High-Order, Singular Finite Elements

For problems with sharp edges, where the electric or magnetic field is singular, p-adaptive finite-element analysis loses some of its effectiveness. This can be remedied by including singular elements which are better able to model the potentials near such edges. The singular elements are hierarchal and the p-adaption takes place over the combined set of singular and regular elements. An energy-based error indicator guides the adaption. Three test cases are considered: an electrostatic problem for finding the capacitance of a stripline gap; a steady current flow problem for finding busbar resistance; and a magnetostatic problem for finding the torque on a prism of permeable material. Results show that, although the use of singular elements does not increase the dimension of the global matrix, more conjugate gradient iterations are needed to solve the matrix equation. Nevertheless, the use of singular elements leads to a greatly improved adaptive convergence, with final accuracies about one order of magnitude higher, for a similar computational cost.