9,940 research outputs found
Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations
We derive a posteriori error estimates for the hybridizable discontinuous
Galerkin (HDG) methods, including both the primal and mixed formulations, for
the approximation of a linear second-order elliptic problem on conforming
simplicial meshes in two and three dimensions.
We obtain fully computable, constant free, a posteriori error bounds on the
broken energy seminorm and the HDG energy (semi)norm of the error. The
estimators are also shown to provide local lower bounds for the HDG energy
(semi)norm of the error up to a constant and a higher-order data oscillation
term. For the primal HDG methods and mixed HDG methods with an appropriate
choice of stabilization parameter, the estimators are also shown to provide a
lower bound for the broken energy seminorm of the error up to a constant and a
higher-order data oscillation term. Numerical examples are given illustrating
the theoretical results
A robust error estimator and a residual-free error indicator for reduced basis methods
The Reduced Basis Method (RBM) is a rigorous model reduction approach for
solving parametrized partial differential equations. It identifies a
low-dimensional subspace for approximation of the parametric solution manifold
that is embedded in high-dimensional space. A reduced order model is
subsequently constructed in this subspace. RBM relies on residual-based error
indicators or {\em a posteriori} error bounds to guide construction of the
reduced solution subspace, to serve as a stopping criteria, and to certify the
resulting surrogate solutions. Unfortunately, it is well-known that the
standard algorithm for residual norm computation suffers from premature
stagnation at the level of the square root of machine precision.
In this paper, we develop two alternatives to the standard offline phase of
reduced basis algorithms. First, we design a robust strategy for computation of
residual error indicators that allows RBM algorithms to enrich the solution
subspace with accuracy beyond root machine precision. Secondly, we propose a
new error indicator based on the Lebesgue function in interpolation theory.
This error indicator does not require computation of residual norms, and
instead only requires the ability to compute the RBM solution. This
residual-free indicator is rigorous in that it bounds the error committed by
the RBM approximation, but up to an uncomputable multiplicative constant.
Because of this, the residual-free indicator is effective in choosing snapshots
during the offline RBM phase, but cannot currently be used to certify error
that the approximation commits. However, it circumvents the need for \textit{a
posteriori} analysis of numerical methods, and therefore can be effective on
problems where such a rigorous estimate is hard to derive
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