Numerical Methods For Nonlinear Elliptic Partial Differential Equations
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Numerical Methods for Nonlinear Elliptic Partial Differential Equations
"The goal of this thesis is to widen the class of provably convergent schemes for elliptic partial differential equations (PDEs) and improve their accuracy. We accomplish this by building on the theory of Barles and Souganidis, and its extension by Froese and Oberman to build monotone and filtered schemes.The first problem considered is the widely studied class of first order Hamilton-Jacobi (HJ) equations. The goal is to construct provably convergent accurate schemes, together with an efficient solver, by making use of the large number of discretizations and solvers already available. To this end, we build filtered schemes, whose main idea is to blend a stable monotone convergent scheme with an accurate scheme while retaining the advantages of both: stability and convergence of the former, and higher accuracy of the latter. Indeed, we are able to build schemes which are second, third, and fourth order accurate in one dimension, as well as schemes that are second order accurate in two dimensions. Moreover, the schemes are explicit, allowing us to use the easy-to-implement fast sweeping method. Using different accurate schemes (e.g. from standard centred differences, higher order upwinding and ENO interpolation), the accuracy of the filtered schemes is validated with computational results for the eikonal equation, as well as other HJ equations (both in one and two dimensions).The second problem considered is the 2-Hessian equation, a fully nonlinear PDE related to the intrinsic curvature for three-dimensional manifolds. The goal is to build numerical methods to compute its solution on a bounded domain given prescribed boundary data. We propose two distinct methods. The first is provably convergent to the unique viscosity solution. The second has higher accuracy and converges in practice, but lacks a formal proof of convergence. The PDE is elliptic on a restricted set of functions: a convexity-type constraint is needed for the ellipticity of the PDE operator, which poses additional difficulties when building the numerical methods. Solutions with both discretizations are obtained using Newton's method. Computational results are presented on a number of exact solutions which range in regularity from smooth to non-differentiable, and in shape from convex to non-convex.The third and last problem is to build a provably convergent scheme for the nonlinear PDE that governs the motion of level sets by affine curvature. It is closely related to mean curvature but exhibits instabilities not found in the former. These instabilities and the lack of regularity of the affine curvature operator posed unexpected and additional difficulties in building monotone schemes. A standard finite difference scheme is proposed and an example that illustrates its nonlinear instability is given. We build provably convergent monotone finite difference schemes. Numerical experiments demonstrate the accuracy and stability of the discretization, as well as the fact that our approximate solutions capture the affine invariance and morphological properties of the evolution." --
Lectures on Numerical Methods for Non-Linear Variational Problems
Author: R. Glowinski
language: en
Publisher: Springer Science & Business Media
Release Date: 2008-01-22
When Herb Keller suggested, more than two years ago, that we update our lectures held at the Tata Institute of Fundamental Research in 1977, and then have it published in the collection Springer Series in Computational Physics, we thought, at first, that it would be an easy task. Actually, we realized very quickly that it would be more complicated than what it seemed at first glance, for several reasons: 1. The first version of Numerical Methods for Nonlinear Variational Problems was, in fact, part of a set of monographs on numerical mat- matics published, in a short span of time, by the Tata Institute of Fun- mental Research in its well-known series Lectures on Mathematics and Physics; as might be expected, the first version systematically used the material of the above monographs, this being particularly true for Lectures on the Finite Element Method by P. G. Ciarlet and Lectures on Optimization—Theory and Algorithms by J. Cea. This second version had to be more self-contained. This necessity led to some minor additions in Chapters I-IV of the original version, and to the introduction of a chapter (namely, Chapter Y of this book) on relaxation methods, since these methods play an important role in various parts of this book.
Numerical Methods for Nonlinear Elliptic Differential Equations
Author: Klaus Böhmer
language: en
Publisher: Oxford University Press
Release Date: 2010-10-07
Boehmer systmatically handles the different numerical methods for nonlinear elliptic problems.