Adaptive higher-order finite element methods for transient PDE problems based on embedded higher-order implicit RungeKutta methods
We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combine adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) RungeKutta methods in time. Weak formulation is only created for the stationary residual, and the RungeKutta methods are specified via their Butchers tables. Around 30 Butchers tables for various RungeKutta methods with numerically verified orders of local and global truncation errors are provided. A time-dependent benchmark problem with known exact solution that contains a sharp moving front is introduced, and it is used to compare the quality of seven embedded implicit higher-order RungeKutta 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 were obtained using the open source library Hermes (http://www.hpfem.org/hermes) and they are reproducible in the Networked Computing Laboratory (NCLab) at http://www.nclab.com.
finite element method rungekutta method butchers table dynamically changing meshes automatic adaptivity reproducible research