A computational tool for comparing all linear PDE solvers - Error-optimal methods are meshless

8 years 14 days ago

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The paper provides a computational technique that allows to compare all linear methods for PDE solving that use the same input data. This is done by writing them as linear recovery formulas for solution values as linear combinations of the input data, and these formulas are continuous linear functionals on Sobolev spaces. Calculating the norm of these functionals on a ﬁxed Sobolev space will then serve as a quality criterion that allows a fair comparison of all linear methods with the same inputs, including standard, extended or generalized ﬁnite–element methods, ﬁnite–difference– and meshless local Petrov–Galerkin techniques. The error bound is computable and yields a sharp worst–case bound in the form of a percentage of the Sobolev norm of the true solution. In this sense, the paper replaces proven error bounds by calculated error bounds. A number of illustrative examples is provided. As a byproduct, it turns out that a unique error–optimal method exists. It necessa...

Robert Schaback

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