Approximating partial differential equations without boundary conditions

TL;DR

This paper introduces a practical near-optimal algorithm for approximating solutions to elliptic PDEs without boundary conditions, using measurement functionals and fractional diffusion techniques to avoid complex boundary inner product computations.

Contribution

It develops a new algorithm that approximates PDE solutions without boundary conditions by leveraging fractional diffusion methods, improving practicality over previous approaches.

Findings

The algorithm achieves near-optimal approximation accuracy.

It avoids complex boundary inner product computations.

The method is applicable to elliptic PDEs with missing boundary data.

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 [Binev et al. 2024]. Inherent to this algorithm is the computation of $H^s$, $s>1/2$, 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 $H^s$ inner products.