Adaptive higher-order finite element methods for transient PDE problems based on embedded higher-order implicit Runge-Kutta methods

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combines adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge-Kutta methods in time. Weak formulation is only created for the stationary residual of the equation, and the Runge-Kutta method is supplied via its Butcher’s table. Around 30 Butcher’s tables for various Runge-Kutta methods with numerically verified orders of local and global truncation errors are provided. New time-dependent benchmark problem with known exact solution that contains a moving front of arbitrary steepness is introduced, and used to compare the quality of seven embedded implicit higher-order Runge-Kutta methods. Numerical experiments also include a comparison of adaptive low-order FEM and hp-FEM with dynamically changing meshes. All numerical results presented in this paper are easily reproducible in the FEMhub Online Lab (http://femhub.org) and using the Hermes open source library (http://hpfem.org/hermes).