Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution effect". Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency

A Continuous hp−hp-Mesh Model for Discontinuous Petrov-Galerkin Finite Element Schemes with Optimal Test Functions

We present an anisotropic $hp-$mesh adaptation strategy using a continuous mesh model for discontinuous Petrov-Galerkin (DPG) finite element schemes with optimal test functions, extending our previous work on $h-$adaptation. The proposed strategy utilizes the inbuilt residual-based error estimator of the DPG discretization to compute both the polynomial distribution and the anisotropy of the mesh elements. In order to predict the optimal order of approximation, we solve local problems on element patches, thus making these computations highly parallelizable. The continuous mesh model is formulated either with respect to the error in the solution, measured in a suitable norm, or with respect to certain admissible target functionals. We demonstrate the performance of the proposed strategy using several numerical examples on triangular grids. Keywords: Discontinuous Petrov-Galerkin, Continuous mesh models, $hp-$ adaptations, Anisotrop