hp-FEM

Summary

The hp-FEM is a version of the finite element method that improves the approximation in two ways at once. It refines the mesh, which makes the element size $h$ smaller, and it raises the polynomial degree $p$ on the elements. Small elements are used near singularities of the solution, and high degrees where the solution is smooth. Take an elliptic problem with piecewise analytic data on a polygon. Babuška and Guo showed that a well chosen hp-sequence then converges exponentially in the number of unknowns. See the definition.