246 research outputs found
The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
The vanishing moment method was introduced by the authors in [37] as a
reliable methodology for computing viscosity solutions of fully nonlinear
second order partial differential equations (PDEs), in particular, using
Galerkin-type numerical methods such as finite element methods, spectral
methods, and discontinuous Galerkin methods, a task which has not been
practicable in the past. The crux of the vanishing moment method is the simple
idea of approximating a fully nonlinear second order PDE by a family
(parametrized by a small parameter \vepsi) of quasilinear higher order (in
particular, fourth order) PDEs. The primary objectives of this book are to
present a detailed convergent analysis for the method in the radial symmetric
case and to carry out a comprehensive finite element numerical analysis for the
vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract
methodological and convergence analysis frameworks of conforming finite element
methods and mixed finite element methods are first developed for fully
nonlinear second order PDEs in general settings. The abstract frameworks are
then applied to three prototypical nonlinear equations, namely, the
Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the
infinity-Laplacian equation. Numerical experiments are also presented for each
problem to validate the theoretical error estimate results and to gauge the
efficiency of the proposed numerical methods and the vanishing moment
methodology.Comment: 141 pages, 16 figure
Stable discontinuous Galerkin FEM without penalty parameters
We propose a modified local discontinuous Galerkin (LDG) method for
second--order elliptic problems that does not require extrinsic penalization to
ensure stability. Stability is instead achieved by showing a discrete
Poincar\'e--Friedrichs inequality for the discrete gradient that employs a
lifting of the jumps with one polynomial degree higher than the scalar
approximation space. Our analysis covers rather general simplicial meshes with
the possibility of hanging nodes.Comment: Accepted for publication in the conference proceedings of Numerical
Mathematics and Advanced Applications ENUMATH 2015. Typo correcte
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