Non-uniform order mixed FEM approximation: implementation, post-processing, computable error bound and adaptivity

The present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form. The work contains: details of the discretisation using non-uniform order mixed finite elements of arbitrarily high order; a new local post-processing scheme for the primary variable; the use of the post-processing scheme in the derivation of new, fully computable bounds for the error in the flux variable; and, an hp-adaptive refinement strategy based on the a posteriori error estimator. Numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten-pole electric motor with singularities arising from both geometric features and discontinuities in material properties. The procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns.