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A local least-squares method for solving nonlinear partial differential equations of second order

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Abstract

In this paper a mesh-free method for the treatment of time-independent and time-dependent nonlinear PDEs of second order is presented. The basic idea of the discretization is a local least-squares approximation, similar to the moving least-squares approach in data approximation. However, in our approach the PDE is incorporated as an additional minimization constraint. The discretization leads to a fixed-point problem, which is solved by iteration. Because of the local nature of the method only small dimensional matrix inversions have to be done. The approximation error of the discretization—even on unstructured meshes—is comparable to respective versions of finite elements. As a by-product the method provides an a posteriori measure for the local approximation error. We discuss implementational aspects and present numerical simulations.

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  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Köln, Germany

    Pascal Heider

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  1. Pascal Heider
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Heider, P. A local least-squares method for solving nonlinear partial differential equations of second order. Numer. Math. 111, 351–375 (2009). https://doi.org/10.1007/s00211-008-0192-4

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  • DOI: https://doi.org/10.1007/s00211-008-0192-4

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