Mapped finite element methods: High-order approximations of problems on domains with cracks and corners

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...

Mortar Finite Volume Element Approximations of Second Order Elliptic Problems

High-order, finite-volume methods in mapped coordinates

We present an approach for constructing finite-volume methods of any order of accuracy for control-volume discretizations of space defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method ...

Enriched Isogeometric Analysis of Elliptic Boundary Value Problems in Domains with Cracks and/or Corners

The mapping method was introduced in [1] for highly accurate Isogeometric Analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push-forwards of B-splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the...

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...