Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A computational tool for comparing all linear PDE solvers -- Optimal
  methods are meshless   The paper starts out with 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. Calculating the norm of these reproduction
formulas 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
finite-element, finite-difference and meshless local Petrov-Galerkin
techniques. A number of illustrative examples will be provided. As a byproduct,
it turns out that a unique error--optimal method exists. It necessarily
outperforms any other competing technique using the same data, e.g. those just
mentioned, and it is necessarily meshless, if solutions are written "entirely
in terms of nodes" (Belytschko et. al. 1996). On closer inspection, it turns
out that it coincides with {\em symmetric meshless collocation} carried out
with the kernel of the Hilbert space used for error evaluation, e.g. with the
kernel of the Sobolev space used. This technique is around since at least 1998,
but its optimality properties went unnoticed, so far.

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The categories of this paper are: astro-ph
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The categories of this paper are: eess
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
3
The categories of this paper are: physics
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The categories of this paper are: gr-qc
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas