Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation

Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution. The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating a Taylor-like series for error in the Lobatto interpolant and using the weak form of the equations yields the correct derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation matrices are given. Computational results illustrate the theory.