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 combine 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, and the Runge–Kutta methods are specified via their Butcher’s tables. Around 30 Butcher’s tables for various Runge–Kutta 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 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 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.