Approximating Partial Differential Equations without Boundary Conditions

Andrea Bonito () and Diane Guignard ()
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Andrea Bonito: Texas A&M University, Department of Mathematics
Diane Guignard: University of Ottawa, Department of Mathematics and Statistics

A chapter in Multiscale, Nonlinear and Adaptive Approximation II, 2024, pp 145-176 from Springer

Abstract: Abstract We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement functionals of the solution. In this context, a near optimal recovery algorithm based on the approximation of the Riesz representers of these functionals in some intermediate Hilbert spaces is proposed and analyzed in [2]. Inherent to this algorithm is the computation of $$ \textit{H}^{\textit{s}} $$ , s > 1/2, inner products on the boundary of the computational domain. We take advantage of techniques borrowed from the analysis of fractional diffusion problems to design and analyze a fully practical near optimal algorithm not relying on the challenging computation of $$ \textit{H}^{\textit{s}} $$ inner products.

Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-75802-7_8

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DOI: 10.1007/978-3-031-75802-7_8

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