1,095 research outputs found
Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian
We generalize the a posteriori techniques for the linear heat equation in [Ver03] to the case of the nonlinear parabolic p-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds
Space-time adaptive finite elements for nonlocal parabolic variational inequalities
This article considers the error analysis of finite element discretizations
and adaptive mesh refinement procedures for nonlocal dynamic contact and
friction, both in the domain and on the boundary. For a large class of
parabolic variational inequalities associated to the fractional Laplacian we
obtain a priori and a posteriori error estimates and study the resulting
space-time adaptive mesh-refinement procedures. Particular emphasis is placed
on mixed formulations, which include the contact forces as a Lagrange
multiplier. Corresponding results are presented for elliptic problems. Our
numerical experiments for -dimensional model problems confirm the
theoretical results: They indicate the efficiency of the a posteriori error
estimates and illustrate the convergence properties of space-time adaptive, as
well as uniform and graded discretizations.Comment: 47 pages, 20 figure
A Posteriori Error Estimation for the p-curl Problem
We derive a posteriori error estimates for a semi-discrete finite element
approximation of a nonlinear eddy current problem arising from applied
superconductivity, known as the -curl problem. In particular, we show the
reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual
type argument and a Helmholtz-Weyl decomposition of
. As a consequence, we are also able to derive an a
posteriori error estimate for a quantity of interest called the AC loss. The
nonlinearity for this form of Maxwell's equation is an analogue of the one
found in the -Laplacian. It is handled without linearizing around the
approximate solution. The non-conformity is dealt by adapting error
decomposition techniques of Carstensen, Hu and Orlando. Geometric
non-conformities also appear because the continuous problem is defined over a
bounded domain while the discrete problem is formulated over a weaker
polyhedral domain. The semi-discrete formulation studied in this paper is often
encountered in commercial codes and is shown to be well-posed. The paper
concludes with numerical results confirming the reliability of the a posteriori
error estimate.Comment: 32 page
A posteriori error control for fully discrete Crank–Nicolson schemes
We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank--Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, which is introduced in the present paper
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