Abstract
Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented auxiliary linear problem (ALP)-based error bounds and corresponding theoretical results, we can prove large improvements in the accuracy of the error predictions compared with existing error bounds. The application of the procedure in parametric model order reduction setting provides a particularly interesting setup, which is why we focus on the application in the reduced basis framework. Several numerical examples illustrate the performance and accuracy of the proposed method.
Highlights
A-posteriori error estimates are important tools in many disciplines of applied mathematics
We show how the idea behind this very simple example can be generalized to a large class of linear and nonlinear problems, especially in the context of reduced basis (RB) methods
To apply the improved error estimation technique, we setup the auxiliary linear problem (ALP), whose weak form in the linear case is given via a(e(μ), v; μ) = a(x(μ), v; μ) − f (v; μ), ∀v ∈ X. Since this equation is as expensive as the original problem, we perform the additional RB approximation of the ALP according to the framework described in the previous section
Summary
A-posteriori error estimates are important tools in many disciplines of applied mathematics. The important question that should be answered is how far the approximation is from the true solution To this end, one typically employs a-posteriori error estimates which in the ideal case deliver a rigorous upper bound that does not deviate too much from the true error. We refer to [3] and [4] for recent overviews of model (order) reduction in the parameteric and nonparametric cases Based on these techniques, approximate solutions can be calculated cheaply and in a computationally efficient manner. We show how the idea behind this very simple example can be generalized to a large class of linear and nonlinear problems, especially in the context of RB methods
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