Kernel-based discretisation for solving matrix-valued PDEs

Giesl, Peter and Wendland, Holger (2018) Kernel-based discretisation for solving matrix-valued PDEs. SIAM Journal on Numerical Analysis, 56 (6). pp. 3386-3406. ISSN 0036-1429

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Abstract

In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems

Item Type: Article
Keywords: Meshfree Methods, Radial Basis Functions, Autonomous Systems, Contraction Metric.
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Subjects: Q Science > QA Mathematics
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Depositing User: Richard Chambers
Date Deposited: 28 Sep 2018 07:23
Last Modified: 17 Jan 2019 08:51
URI: http://sro.sussex.ac.uk/id/eprint/79058

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