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Displaying 781 –
800 of
9187
We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput. 75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using...
We prove a posteriori error estimates of optimal order for linear
Schrödinger-type equations in the L∞(L2)- and the
L∞(H1)-norm. We discretize only in time by the
Crank-Nicolson method. The direct use of the reconstruction
technique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds that
are of optimal order in the L∞(L2)-norm, but of
suboptimal order in the L∞(H1)-norm. The optimality in
the case of L∞(H1)-norm is recovered by using...
We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say , is governed by an elliptic equation and the other, say , by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the - and -components to obtain optimally convergent a priori bounds for all the terms in the error energy norm....
We analyze Euler-Galerkin approximations (conforming finite elements in
space and implicit Euler in time) to
coupled PDE systems in which one dependent
variable, say u, is governed by an elliptic equation and the other,
say p, by a parabolic-like equation. The underlying application is the
poroelasticity system within the quasi-static assumption. Different
polynomial orders are used for the u- and p-components to
obtain optimally convergent a priori bounds for all
the terms in the error energy...
The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space spanned by solutions of the...
In this paper, we extend the reduced-basis methods and associated a
posteriori error estimators developed earlier for elliptic partial
differential equations to parabolic problems with affine parameter
dependence. The essential new ingredient is the presence of time in the
formulation and solution of the problem – we shall “simply” treat
time as an additional, albeit special, parameter. First, we introduce
the reduced-basis recipe – Galerkin projection onto a space WN
spanned by solutions...
Phase-field models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on . Using an energy argument combined with a topological continuation argument and a spectral...
Phase-field models, the simplest of which is Allen–Cahn's
problem, are characterized by a small parameter ε that dictates
the interface thickness. These models naturally call for mesh adaptation
techniques, which rely on a posteriori error control.
However, their error analysis usually deals with the
underlying non-monotone nonlinearity via a Gronwall argument which
leads to an exponential dependence on ε-2. Using an energy argument
combined with a
topological continuation argument and...
We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...
The paper presents an a posteriori error estimator for a (piecewise linear)
nonconforming finite element approximation of the heat equation
in , d=2 or 3,
using backward Euler's scheme.
For this discretization, we derive a residual indicator, which use
a spatial residual indicator based on the
jumps of normal and tangential derivatives of the nonconforming
approximation and
a time residual indicator based on the jump of broken gradients at each time step.
Lower and
upper bounds form the main...
In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation...
We present new a posteriori error estimates for the finite volume approximations
of elliptic problems. They are obtained by applying functional a posteriori
error estimates to natural extensions of the approximate solution and its flux
computed by the finite volume method. The estimates give guaranteed upper bounds
for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and
also in terms of the combined primal-dual norms. It is shown that the estimates
provide sharp upper and...
In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping
(given in terms of a boundary integral operator) to solve linear exterior transmission problems in
the plane. As a model we consider a second order elliptic equation in divergence form coupled with
the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational
formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive
the...
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