Jens Markus Melenk

The partition of unity finite element method: Basic theory and applications

The paper presents the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM). We will show how the PUFEM can be used to employ the structure of the differential equation under consideration to construct effective and robust methods. Although the method and its theory are valid in n dimensions, a detailed and illustrative analysis will be given for a one-dimensional model problem. We identify some classes of non-standard problems which can profit highly from the advantages of the PUFEM and conclude this paper with some open questions concerning implementational aspects of the PUFEM.

THE PARTITION OF UNITY METHOD

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a posteriori error estimation for this new method are also proved.

On residual-based a posteriori error estimation in hp -FEM

A family ηα, α∈[0,1], of residual-based error indicators for the hp-version of the finite element method is presented and analyzed. Upper and lower bounds for the error indicators ηα are established. To do so, the well-known Clément/Scott–Zhang interpolation operator is generalized to the hp-context and new polynomial inverse estimates are presented. An hp-adaptive strategy is proposed. Numerical examples illustrate the performance of the error indicators and the adaptive strategy.

HP-finite element methods for singular perturbations

1.Introduction.- Part I: Finite Element Approximation.- 2. hp-FEM for Reaction Diffusion Problems: Principal Results.- 3. hp Approximation.- Part II: Regularity in Countably Normed Spaces.- 4. The Countably Normed Spaces blb,e.- 5. Regularity Theory in Countably Normed Spaces.- Part III: Regularity in Terms of Asymptotic Expansions.- 6. Exponentially Weighted Countably Normed Spaces.- Appendix.- References.- Index.

Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in

Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation

\Bbb R^d, d \in \{1,2,3\}

Fully discrete hp-finite elements : fast quadrature

is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical

hp -Interpolation of Nonsmooth Functions and an Application to hp -A posteriori Error Estimation

hp

On Stability of Discretizations of the Helmholtz Equation

-version of the finite element method is presented for the model problem where the dependence on the mesh width

hp FEM for Reaction-Diffusion Equations I: Robust Exponential Convergence

h,