Adaptive Algorithms

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations

High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $W^1_\infty$ and piecewise in a suitable Besov class embedded in $C^{1,\alpha}$ with $\alpha \in (0,1]$. The idea is to have the surface sufficiently well resolved in $W^1_\infty$ relative to the current resolution of the PDE in $H^1$. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in $W^1_\infty$ and PDE error in $H^1$.Comment: 51 pages, the published version contains an additional glossar

Adaptive Numerical Methods for PDEs

This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods

Convergence of an adaptive mixed finite element method for general second order linear elliptic problems

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem. An adaptive marking in each step for the local refinement is based on the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments confirm the theoretical analysis.Comment: 24 pages, 8 figure

Adaptive finite elements for viscoelastic deformation problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis is concerned with the theoretical and computational aspects of generating solutions to problems involving materials with fading memory, known as viscoelastic materials. Viscoelastic materials can be loosely described as those whose current stress configuration depends on their recent past. Viscoelastic constitutive laws for stress typically take the form of a sum of an instantaneous response term and an integral over their past responses. Such laws are called hereditary integral constitutive laws. The main purpose of this study is to analyse adaptive finite element algorithms for the numerical solution of the quasistatic equations governing the small displacement of a viscoelastic body subjected to prescribed body forces and tractions. Such algorithms for the hereditary integral formulation have appeared in the literature. However the approach here is to consider an equivalent formulation based on the introduction of a set of unobservable interval variables. In the linear viscoelastic case we exploit the structure of the quasistatic problem to remove the displacement from the equations governing the internal variables. This results in an elliptic problem with right hand side dependent on the internal variables, and a separate independent system of ordinary differential equations in a Hilbert space. We consider a continuous in space and time Galerkin finite element approximation to the reformulated problem for which we derive optimal order a priori error estimates. We then apply the techniques of the theory of adaptive finite element methods for elliptic boundary value problems and ordinary differential equations, deriving reliable and efficient a posteriori error estimates and detailing adaptive algorithms. We consider the idea of splitting the error into space and time portions and present results regarding a splitting for space time projections. The ideas for splitting the error in projections is applied to the finite element approximation and a further set of a posteriori error estimates derived. Numerical studies confirm the theoretical properties of all of the estimators and we show how they can be used to drive adaptive in space and time solution algorithms. We consider the extension of our results for the linear case to the constitutively nonlinear case. A model problem is formulated and the general techniques for dealing with a posterior error estimation for nonlinear space time problems are considered.EPSRC; Japanese Society for the Promotion of Science(JSPS

Adaptive SDE based interpolation for random PDEs

A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method