PDE-independent adaptive hphp-FEM based on hierarchic extension of finite element spaces

We present a novel approach to automatic adaptivity in higher-order finite element methods (hphp-FEM) which is free of analytical error estimates. This means that a computer code based on this approach can be used to solve adaptively a wide range of PDE problems. A posteriori error estimation is done computationally via hierarchic extension of finite element spaces. This is an analogy to embedded higher-order methods for ODE. The adaptivity process yields a sequence of embedded stiffness matrices which are solved efficiently using a simple combined direct-iterative algorithm. The methodology works equally well for standard low-order FEM and for the hphp-FEM. Numerical examples are presented.