Petr Knobloch

Non-nested multi-level solvers for finite element discretisations of mixed problems

Abstract We consider a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type. As a basic new feature, our approach allows to use different finite element discretisations on each level of the multi-grid hierarchy. Thus, in our multi-level approach, accurate higher order finite element discretisations can be combined with fast multi-level solvers based on lower order (nonconforming) finite element discretisations. This leads to the design of efficient multi-level solvers for higher order finite element discretisations.

A Generalization of the Local Projection Stabilization for Convection-Diffusion-Reaction Equations

We introduce a generalization of the local projection stabilization for steady scalar convection-diffusion-reaction equations which allows us to use local projection spaces defined on overlapping sets. This enables us to define the local projection method without the need of a mesh refinement or an enrichment of the finite element space and increases the robustness of the local projection method with respect to the choice of the stabilization parameter. The stabilization term is slightly modified, which leads to an optimal estimate of the consistency error even if the stabilization parameters scale correctly with respect to convection, diffusion, and mesh width. We prove that the bilinear form corresponding to the method satisfies an inf-sup condition with respect to the SUPG norm and establish an optimal error estimate in this norm. The theoretical considerations are illustrated by numerical results.

The

We consider a nonconforming streamline diffusion finite element method for solving convection-diffusion problems. The loss of the Galerkin orthogonality of the streamline diffusion method when applied to nonconforming finite element approximations results in an additional error term which cannot be estimated uniformly with respect to the perturbation parameter for the standard piecewise linear or rotated bilinear elements. Therefore, starting from the Crouzeix--Raviart element, we construct a modified nonconforming first order finite element space on shape regular triangular meshes satisfying a patch test of higher order. A rigorous error analysis of this

P_1^\protect\lowercasemod

P_1^{\mbox{\scriptsize\it mod}}

Element: A New Nonconforming Finite Element for Convection-Diffusion Problems

element applied to a streamline diffusion discretization is given. The numerical tests show the robustness and the high accuracy of the new method.

On the performance of SOLD methods for convection diffusion problems with interior layers

Numerical solutions of convection diffusion equations obtained using the Streamline Upwind Petrov Galerkin (SUPG) stabilisation typically possess spurious oscillations at layers. Spurious Oscillations at Layers Diminishing (SOLD) methods aim to suppress or at least diminish these oscillations without smearing the layers extensively. In the recent review by John and Knobloch (2007), numerical studies at convection diffusion problems with constant convection whose solutions have boundary layers led to a pre-selection of the best available SOLD methods with respect to the two goals stated above. The behaviour of these methods is studied in this paper for a convection diffusion problem with a non-constant convection field whose solution possesses an interior layer.

A comparison of spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. - Part I

An unwelcome feature of the popular streamline upwind/Petrov-Galerkin (SUPG) stabilization of convection-dominated convection-diffusion equations is the presence of spurious oscillations at layers. Since the mid of the 1980-ies, a number of methods have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers. The paper gives a review and state of the art of these methods, discusses their derivation, proposes some alternative choices of parameters in the methods and categorizes them. Some numerical studies give a first insight into the advantages and drawbacks of the methods.

On the choice of the SUPG parameter at outflow boundary layers

We consider the Streamline upwind/Petrov–Galerkin (SUPG) finite element method for two–dimensional steady scalar convection–diffusion equations and propose a new definition of the SUPG stabilization parameter along outflow Dirichlet boundaries. Numerical results demonstrate a significant improvement of the accuracy and show that, in some cases, even nodally exact solutions are obtained.

On the Application of Local Projection Methods to Convection–Diffusion–Reaction Problems

We apply the local projection stabilization to finite element discretizations of scalar convection-diffusion-reaction equations with mixed boundary conditions. We derive general error estimates and discuss the choice of the stabilization parameter. Numerical results illustrate some drawbacks of the local projection stabilization in comparison to the SUPG method.

An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes

This work is devoted to the proposal of a new flux limiter that makes the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes. Minimal assumptions on the limiter are given in order to guarantee the validity of the discrete maximum principle, and then a precise definition of it is proposed and analyzed. Numerical results for convection–diffusion problems confirm the theory.