Work

Solving the unique continuation problem for Schrödinger equations with low regularity solutions using a stabilized finite element method

Year: 2024

Type: preprint

Abstract: In this paper, we consider the unique continuation problem for the Schrödinger equations. We prove a Hölder type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a priori} knowledge of the solution, achieving error estimates in interior domains with convergence up to continuous stability. The approximability of the scheme to solutions with only $H^1$-regularity is studied and the convergence rate for solutions with regularity higher than $H^1$ is also shown. Comparisons in terms of different parameterization for different regularities will be illustrated with respect to the convergence and condition numbers of the linear systems. Finally, numerical experiments will be given to illustrate the theory.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2403.16914

Authors: Burman, Erik, Lu, Mingfei, Oksanen, Lauri

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funders: European Commission, Engineering and Physical Sciences Research Council

Awards: Quantitative estimates of discretisation and modelling errors in variational data assimilation for incompressible flows, Computational methods for inverse problems subject to wave equations in heterogeneous media, G8144583337

OpenAlex ID: W4393213845

Cites:

Cited by:

Related to: 10