Journal ArticleDOI
Generation of Difference and Error Formulae of Arbitrary Consistency Order on an Unstructured Grid
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In this article, the problem of finding grid points with sufficient information for the determination of the interpolation polynominals, good error estimate and optimal structure of the resulting linear system of equations of the FDM is presented.Abstract:
For the application of the finite difference method (FDM) on an unstructured finite element grid we want to generate difference formulae and corresponding error estimates of arbitrary consistency order. The problem is to find grid points with sufficient information for the determination of the interpolation polynominals, good error estimate and optimal structure of the resulting linear system of equations of the FDM. A sophisticated algorithm for the selection of good and the exclusion of bad grid points is presented. For the generation of the formulae the principle of the influence polynomials is used.read more
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Journal ArticleDOI
How We solve PDEs
Willi Schönauer,Torsten Adolph +1 more
TL;DR: The FDEM program package will be a black-box solver for nonlinear systems of elliptic and parabolic PDEs with mesh refinement and automatic control of the consistency order in each space grid point.
Journal ArticleDOI
Meshfree collocation solution of boundary value problems via interpolating moving least squares
TL;DR: In this article, a mesh-free interpolating moving least squares (IMLS) method based on singular weights for the solution of partial differential equations is presented, where the singular kernel functions lead to really interpolating functions which satisfy the Kronecker-delta property.
Journal ArticleDOI
A unified algorithm for the selection of collocation stencils for convex, concave and singular problems
TL;DR: A unified algorithm is introduced which allows the selection of collocation stencils, based on the visibility criterion, for convex, concave, and singular problems solved using a collocation method, and is particularly useful in the context of model refinement.
Journal ArticleDOI
Difference operators from interpolating moving least squares and their deviation from optimality
TL;DR: In this article, the first and second derivative of the IMLS interpolant are computed at the nodes of a given grid with the help of a basic lemma on Shepard interpolants, and the difference formulae are compared with those defining optimal finite difference methods and discuss their deviation from optimality.
Book
A New Meshless Collocation Method for Partial Differential Equations
TL;DR: In this article, a collocation meshless method is developed for the numerical solution of partial differential equations (PDEs) on the scattered point distribution, where meshless shape functions are constructed on a group of selected stencils arbitrarily distributed in a local support domain by means of a polynomial interpolation.