7 research outputs found
A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems
We use the elliptic reconstruction technique in combination with a duality
approach to prove aposteriori error estimates for fully discrete back- ward
Euler scheme for linear parabolic equations. As an application, we com- bine
our result with the residual based estimators from the aposteriori esti- mation
for elliptic problems to derive space-error indicators and thus a fully
practical version of the estimators bounding the error in the L \infty (0, T ;
L2({\Omega})) norm. These estimators, which are of optimal order, extend those
introduced by Eriksson and Johnson (1991) by taking into account the error
induced by the mesh changes and allowing for a more flexible use of the
elliptic estima- tors. For comparison with previous results we derive also an
energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error
which simplifies a previous one given in Lakkis and Makridakis (2006). We then
compare both estimators (duality vs. energy) in practical situations and draw
conclusions.Comment: 30 pages, including 7 color plates in 4 figure
The Postprocessed Mixed Finite-Element Method for the Navier--Stokes Equations
A postprocessing technique for mixed finite-element methods for the incompressible
Navier鈥揝tokes equations is studied. The technique was earlier developed for spectral and standard
finite-element methods for dissipative partial differential equations. The postprocessing amounts to
solving a Stokes problem on a finer grid (or higher-order space) once the time integration on the
coarser mesh is completed. The analysis presented here shows that this technique increases the
convergence rate of both the velocity and the pressure approximations. Numerical experiments are
presented that confirm both this increase in the convergence rate and the corresponding improvement
in computational efficiency.DGICYT BFM2003-0033
A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems
We use the elliptic reconstruction technique in combination with a duality
approach to prove aposteriori error estimates for fully discrete back- ward
Euler scheme for linear parabolic equations. As an application, we com- bine
our result with the residual based estimators from the aposteriori esti- mation
for elliptic problems to derive space-error indicators and thus a fully
practical version of the estimators bounding the error in the L \infty (0, T ;
L2({\Omega})) norm. These estimators, which are of optimal order, extend those
introduced by Eriksson and Johnson (1991) by taking into account the error
induced by the mesh changes and allowing for a more flexible use of the
elliptic estima- tors. For comparison with previous results we derive also an
energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error
which simplifies a previous one given in Lakkis and Makridakis (2006). We then
compare both estimators (duality vs. energy) in practical situations and draw
conclusions.Comment: 30 pages, including 7 color plates in 4 figure
Postprocessing finite-element methods for the Navier鈥揝tokes Equations: the Fully discrete case
An accuracy-enhancing postprocessing technique for finite-element discretizations
of the Navier鈥揝tokes equations is analyzed. The technique had been previously analyzed only for
semidiscretizations, and fully discrete methods are addressed in the present paper. We show that
the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising
from any convergent time-stepping procedure. Further refined bounds are obtained when the timestepping
procedure is either the backward Euler method or the two-step backward differentiation
formula
The Postprocessed Mixed Finite-Element Method for the Navier鈥揝tokes Equations: Refined Error Bounds
A postprocessing technique for mixed finite-element methods for the incompressible Navier鈥揝tokes equations is analyzed. The postprocess, which amounts to solving a (linear) Stokes problem, is shown to increase the order of convergence of the method to which it is applied by one unit (times the logarithm of the mesh diameter). In proving the error bounds, some superconvergence results are also obtained. Contrary to previous analysis of the postprocessing technique, in the present paper we take into account the loss of regularity suffered by the solutions of the Navier鈥揝tokes equations at the initial time in the absence of nonlocal compatibility conditions of the data.Ministerio de Educaci贸n y Ciencia MTM2006- 0084
Guaranteed Verification of Finite Element Solutions of Heat Conduction
This dissertation addresses the accuracy of a-posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather
coarse meshes are used, which are often encountered in engineering computations. We present sample computations which indicate lack of robustness of all standard
residual estimators with respect to high orthotropy. The investigation shows that the main culprit behind the lack of robustness of residual estimators is the coarseness
of the finite element meshes relative to the thickness of the boundary and interface layers in the solution.
With the introduction of an elliptic reconstruction procedure, a new error estimator based on the solution of the elliptic reconstruction problem is invented to
estimate the exact error measured in space-time C-norm for both semi-discrete and fully discrete finite element solutions to linear parabolic problem. For a fully discrete solution, a temporal error estimator is also introduced to evaluate the discretization error in the temporal field. In the meantime, the implicit Neumann subdomain residual estimator for elliptic equations, which involves the solution of the local residual
problem, is combined with the elliptic reconstruction procedure to carry out a posteriori error estimation for the linear parabolic problem. Numerical examples are
presented to illustrate the superconvergence properties in the elliptic reconstruction and the performance of the bounds based on the space-time C-norm.
The results show that in the case of L^2 norm for smooth solution there is no superconvergence in elliptic reconstruction for linear element, and for singular solution the superconvergence does not exist for element of any order while in the case of energy norm the superconvergence always exists in elliptic reconstruction. The research also shows that the performance of the bounds based on space-time C-norm is robust, and in the case of fully discrete finite element solution the bounds for the temporal error are sharp