Robust high-order unfitted finite elements by interpolation-based discrete extension

Analysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method

The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation n...

Discrete maximum principle for higher-order finite elements in 1D

We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the hp-FEM. The DMP holds if a relative length of every element K in the mesh is bounded by a value H∗ rel(p) ∈ [0.9, 1], where p ≥ 1 is the polynomial degree of the element K. The values H∗ rel(p) ar...

New robust nonconforming finite elements of higher order

We study second order nonconforming finite elements as members of a new family of higher order approaches which behave optimally not only on multilevel refined grids, but also on perturbed grids which are still shape regular but which consist no longer of asymptotically affine equivalent mesh cells. We present two approaches to prevent this order reduction: The first one is based on the use of ...

High order unfitted finite element methods on level set domains using isoparametric mappings

We introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions. An unfitted finite element method which is suitable for the case of piecewise planar interfaces is combined with a parametric mapping of the underlying mesh resulting in an isoparametric unfitted fin...

High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces