354 research outputs found
Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces
Convergence results are shown for full discretizations of quasilinear
parabolic partial differential equations on evolving surfaces. As a
semidiscretization in space the evolving surface finite element method is
considered, using a regularity result of a generalized Ritz map, optimal order
error estimates for the spatial discretization is shown. Combining this with
the stability results for Runge--Kutta and BDF time integrators, we obtain
convergence results for the fully discrete problems.Comment: -. arXiv admin note: text overlap with arXiv:1410.048
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
Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization
We study a doubly nonlinear parabolic problem arising in the modeling of gas
transport in pipelines. Using convexity arguments and relative entropy
estimates we show uniform bounds and exponential stability of discrete
approximations obtained by a finite element method and implicit time stepping.
Due to convergence of the approximations to weak solutions of the problem, our
results also imply regularity, uniqueness, and long time stability of weak
solutions of the continuous problem
FINITE-ELEMENT APPROXIMATION OF THE PARABOLIC P-LAPLACIAN
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