5,665 research outputs found
Recovery-Based Error Estimators for Diffusion Problems: Explicit Formulas
We introduced and analyzed robust recovery-based a posteriori error
estimators for various lower order finite element approximations to interface
problems in [9, 10], where the recoveries of the flux and/or gradient are
implicit (i.e., requiring solutions of global problems with mass matrices). In
this paper, we develop fully explicit recovery-based error estimators for lower
order conforming, mixed, and non- conforming finite element approximations to
diffusion problems with full coefficient tensor. When the diffusion coefficient
is piecewise constant scalar and its distribution is local quasi-monotone, it
is shown theoretically that the estimators developed in this paper are robust
with respect to the size of jumps. Numerical experiments are also performed to
support the theoretical results
Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations
The first-order system LL* (FOSLL*) approach for general second-order
elliptic partial differential equations was proposed and analyzed in [10], in
order to retain the full efficiency of the L2 norm first-order system
least-squares (FOSLS) ap- proach while exhibiting the generality of the
inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the
div-curl system with added slack vari- ables, and hence it is quite
complicated. In this paper, we apply the FOSLL* approach to the div system and
establish its well-posedness. For the corresponding finite ele- ment
approximation, we obtain a quasi-optimal a priori error bound under the same
regularity assumption as the standard Galerkin method, but without the
restriction to sufficiently small mesh size. Unlike the FOSLS approach, the
FOSLL* approach does not have a free a posteriori error estimator, we then
propose an explicit residual error estimator and establish its reliability and
efficiency bound
Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements
In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu
(ZZ) estimator for the conforming linear finite element approximation to
elliptic interface problems. The estimator is based on the piecewise "constant"
flux recovery in the conforming finite element space. This
paper extends the results of \cite{CaZh:09} to diffusion problems with full
diffusion tensor and to the flux recovery both in piecewise constant and
piecewise linear space.Comment: arXiv admin note: substantial text overlap with arXiv:1407.437
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