Summary
Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called “Mehrstellenformeln’ or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.
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Yserentant, H. Die mehrstellenformeln für den Laplaceoperator. Numer. Math. 34, 171–187 (1980). https://doi.org/10.1007/BF01396058
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DOI: https://doi.org/10.1007/BF01396058