Finite Element Approximation of Some Degenerate/Singular Elliptic And Parabolic Equations
File(s)
Author(s)
Schneider, Mathias
Type
Thesis
Abstract
The main purpose of this thesis is to prove a priori error bounds for the standard continuous piecewise linear finite element approximation of a semilinear elliptic problem with a singular nonlinearity which occurs in nonlinear elasticity and a semilinear degenerate parabolic problem modelling fast diffusion.
The elliptic problem can be stated as follows: Given z/ G R+, find u > 0 such that
—Aiz = cu~^ in Q , u = Q on ,
where Q C with sufficiently smooth boundary, d = 1, 2 or 3, and c > 0 in Q. Although this problem has been studied in the literature from an analytical point of view, we are not aware of any finite element error analysis on it. We prove and error bounds for the standard Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and c G R+ we analyse a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal and error bounds for d = 1. For this case we also present some numerical results.
The parabolic problem modelling fast diffusion is to find u such that
+ / in Qt := Q X (0, T] , u = Q on X (0, T] ,
0(w)(-,O) = in Q ,
where Q is convex polyhedral in d = 1, 2 or 3, ^(5) := A sign(s) |sp with A > 0 and p > 1 and f, g given. There is a vast literature on the singular (i.e. p < 1} form of the given problem, which models slow diffusion. Considerably less work has been done on the degenerate case, p > 1. We are aware of only one article which proves an error bound for a fully practical Galerkin scheme; and this only for d = 1.
We prove error bounds for the standard Galerkin finite element approximation in space and backward Euler time discretization. Our bounds hold for d = 1,2 and 3. In addition, we analyse a more practical scheme involving numerical integration in space on the nonlinear terms and derive error bounds for the fully practical method. Our bounds improve on those in the literature both in terms of the order of convergence and in terms of the size of the time step required for the computation. After that we include an adaptation to our setting of a novel but abstract approach given by Rulla for the analysis of the backward Euler method.
We also deal with the finite time extinction property which distinguishes the fast from the slow diffusion equation and its appropriate numerical treatment via Le Roux’s method. Numerical results for both the backward Euler and Le Roux’s method are presented for d — 1. We conclude this thesis with a brief look at linear schemes deriving from extrapolation and relaxation.
The elliptic problem can be stated as follows: Given z/ G R+, find u > 0 such that
—Aiz = cu~^ in Q , u = Q on ,
where Q C with sufficiently smooth boundary, d = 1, 2 or 3, and c > 0 in Q. Although this problem has been studied in the literature from an analytical point of view, we are not aware of any finite element error analysis on it. We prove and error bounds for the standard Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and c G R+ we analyse a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal and error bounds for d = 1. For this case we also present some numerical results.
The parabolic problem modelling fast diffusion is to find u such that
+ / in Qt := Q X (0, T] , u = Q on X (0, T] ,
0(w)(-,O) = in Q ,
where Q is convex polyhedral in d = 1, 2 or 3, ^(5) := A sign(s) |sp with A > 0 and p > 1 and f, g given. There is a vast literature on the singular (i.e. p < 1} form of the given problem, which models slow diffusion. Considerably less work has been done on the degenerate case, p > 1. We are aware of only one article which proves an error bound for a fully practical Galerkin scheme; and this only for d = 1.
We prove error bounds for the standard Galerkin finite element approximation in space and backward Euler time discretization. Our bounds hold for d = 1,2 and 3. In addition, we analyse a more practical scheme involving numerical integration in space on the nonlinear terms and derive error bounds for the fully practical method. Our bounds improve on those in the literature both in terms of the order of convergence and in terms of the size of the time step required for the computation. After that we include an adaptation to our setting of a novel but abstract approach given by Rulla for the analysis of the backward Euler method.
We also deal with the finite time extinction property which distinguishes the fast from the slow diffusion equation and its appropriate numerical treatment via Le Roux’s method. Numerical results for both the backward Euler and Le Roux’s method are presented for d — 1. We conclude this thesis with a brief look at linear schemes deriving from extrapolation and relaxation.
Version
Open Access
Date Issued
1997
Date Awarded
1997-05
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Barrett, John
Sponsor
Engineering and Physical Sciences Research Council, U.K.; Studienstiftung des Deutschen Volkes, Germany; Harry Jones Scholarship through the Applied Mathematics/Numerical Analysis Section of the Maths Department at Imperial.
Publisher Department
Department of Mathematics
Publisher Institution
University of London - Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)