Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Notes on error estimates for the standard Galerkin-finite element method
  for the Shallow Water equations   We consider a simple initial-boundary-value problem for the shallow water
equations in one space dimension, and also the analogous problem for a
symmetric variant of the system. Assuming smoothness of solutions, we
discretize these problems in space using standard Galerkin-finite element
methods and prove $L^{2}$-error estimates for the semidiscrete problems for
quasiuniform and uniform meshes. In particular we show that in the case of
spatial discretizations with piecewise linear continuous functions on a uniform
mesh, suitable compatibility conditions at the boundary and superaccuracy
properties of the $L^{2}$ projection on the finite element subspaces lead to an
optimal-order $O(h^{2})$ $L^{2}$-error estimate. We also examine temporal
discretizations of the semidiscrete problems by three explicit Runge-Kutta
methods (the Euler, improved Euler, and the Shu-Osher scheme) and prove
$L^{2}$-error estimates, which are of optimal order in the temporal variable,
under appropriate stability conditions. In a final section of remarks we prove
optimal-order $L^{2}$-error estimates for smooth spline spatial discretizations
of the periodic initial-value problem for the systems. We also prove that
small-amplitude, appropriately transformed solutions of the symmetric system
are close to the corresponding solutions of the usual system while they are
both smooth, thus providing a justification of the symmetric system.

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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph