830 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
Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows
In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm
An hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type
In this paper, an hp-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp-quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H1 norm and L2 norm which are optimal in h, suboptimal in p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
A posteriori error control for discontinuous Galerkin methods for parabolic problems
We derive energy-norm a posteriori error bounds for an Euler time-stepping
method combined with various spatial discontinuous Galerkin schemes for linear
parabolic problems. For accessibility, we address first the spatially
semidiscrete case, and then move to the fully discrete scheme by introducing
the implicit Euler time-stepping. All results are presented in an abstract
setting and then illustrated with particular applications. This enables the
error bounds to hold for a variety of discontinuous Galerkin methods, provided
that energy-norm a posteriori error bounds for the corresponding elliptic
problem are available. To illustrate the method, we apply it to the interior
penalty discontinuous Galerkin method, which requires the derivation of novel a
posteriori error bounds. For the analysis of the time-dependent problems we use
the elliptic reconstruction technique and we deal with the nonconforming part
of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure
- …