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

作者：Solin Pavel^*; Korous Lukas

来源：Journal of Computational Physics, 2012, 231(4): 1635-1649.

DOI：10.1016/j.jcp.2011.10.023

• Runge-Kutta method
• Butcher%26apos
• s table
• Finite element method
• Automatic adaptivity
• Dynamically changing meshes
• Reproducible research

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%26apos;s tables. Around 30 Butcher%26apos;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.

• 出版日期2012-2-20