1,146 research outputs found
Functional a posteriori error estimates for time-periodic parabolic optimal control problems
This paper is devoted to the a posteriori error analysis of multiharmonic
finite element approximations to distributed optimal control problems with
time-periodic state equations of parabolic type. We derive a posteriori
estimates of functional type, which are easily computable and provide
guaranteed upper bounds for the state and co-state errors as well as for the
cost functional. These theoretical results are confirmed by several numerical
tests that show high efficiency of the a posteriori error bounds
Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation
This paper presents an a-posteriori goal-oriented error analysis for a
numerical approximation of the steady Boltzmann equation based on a
moment-system approximation in velocity dependence and a discontinuous Galerkin
finite-element (DGFE) approximation in position dependence. We derive
computable error estimates and bounds for general target functionals of
solutions of the steady Boltzmann equation based on the DGFE moment
approximation. The a-posteriori error estimates and bounds are used to guide a
model adaptive algorithm for optimal approximations of the goal functional in
question. We present results for one-dimensional heat transfer and shock
structure problems where the moment model order is refined locally in space for
optimal approximation of the heat flux.Comment: arXiv admin note: text overlap with arXiv:1602.0131
Functional a posteriori error estimates for parabolic time-periodic boundary value problems
The paper is concerned with parabolic time-periodic boundary value problems
which are of theoretical interest and arise in different practical
applications. The multiharmonic finite element method is well adapted to this
class of parabolic problems. We study properties of multiharmonic
approximations and derive guaranteed and fully computable bounds of
approximation errors. For this purpose, we use the functional a posteriori
error estimation techniques earlier introduced by S. Repin. Numerical tests
confirm the efficiency of the a posteriori error bounds derived
A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws
We present reliable a posteriori estimators for some fully discrete schemes
applied to nonlinear systems of hyperbolic conservation laws in one space
dimension with strictly convex entropy. The schemes are based on a method of
lines approach combining discontinuous Galerkin spatial discretization with
single- or multi-step methods in time. The construction of the estimators
requires a reconstruction in time for which we present a very general framework
first for odes and then apply the approach to conservation laws. The
reconstruction does not depend on the actual method used for evolving the
solution in time. Most importantly it covers in addition to implicit methods
also the wide range of explicit methods typically used to solve conservation
laws. For the spatial discretization, we allow for standard choices of
numerical fluxes. We use reconstructions of the discrete solution together with
the relative entropy stability framework, which leads to error control in the
case of smooth solutions. We study under which conditions on the numerical flux
the estimate is of optimal order pre-shock. While the estimator we derive is
computable and valid post-shock for fixed meshsize, it will blow up as the
meshsize tends to zero. This is due to a breakdown of the relative entropy
framework when discontinuities develop. We conclude with some numerical
benchmarking to test the robustness of the derived estimator
Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions
We prove strong convergence of conforming finite element approximations to
the stationary Joule heating problem with mixed boundary conditions on
Lipschitz domains in three spatial dimensions. We show optimal global
regularity estimates on creased domains and prove a priori and a posteriori
bounds for shape regular meshes.Comment: Keywords: Joule heating problem, thermistors, a posteriori error
analysis, a priori error analysis, finite element metho
A posteriori analysis of discontinuous galerkin schemes for systems of hyperbolic conservation laws
In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator
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