1,078 research outputs found
A posteriori error control for discontinuous Galerkin methods for parabolic problems
We derive energy-norm a posteriori error bounds for an Euler time-stepping
method combined with various spatial discontinuous Galerkin schemes for linear
parabolic problems. For accessibility, we address first the spatially
semidiscrete case, and then move to the fully discrete scheme by introducing
the implicit Euler time-stepping. All results are presented in an abstract
setting and then illustrated with particular applications. This enables the
error bounds to hold for a variety of discontinuous Galerkin methods, provided
that energy-norm a posteriori error bounds for the corresponding elliptic
problem are available. To illustrate the method, we apply it to the interior
penalty discontinuous Galerkin method, which requires the derivation of novel a
posteriori error bounds. For the analysis of the time-dependent problems we use
the elliptic reconstruction technique and we deal with the nonconforming part
of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
In this paper, we analyze the convergence and optimality of a standard
adaptive nonconforming linear element method for the Stokes problem. After
establishing a special quasi--orthogonality property for both the velocity and
the pressure in this saddle point problem, we introduce a new prolongation
operator to carry through the discrete reliability analysis for the error
estimator. We then use a specially defined interpolation operator to prove
that, up to oscillation, the error can be bounded by the approximation error
within a properly defined nonlinear approximate class. Finally, by introducing
a new parameter-dependent error estimator, we prove the convergence and
optimality estimates
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