Conforming and Nonconforming Finite Element Methods for Biharmonic Inverse Source Problem

TL;DR

This paper develops a unified framework for analyzing conforming and nonconforming finite element methods applied to the biharmonic inverse source problem, including error estimates and regularisation techniques, supported by numerical validation.

Contribution

It introduces a comprehensive analysis of conforming and Morley nonconforming FEMs for the biharmonic inverse problem, including error estimates and regularisation strategies.

Findings

Error estimates for forward solutions in abstract setting

Error bounds for regularised inverse solutions

Numerical results confirming theoretical predictions

Abstract

This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularisation is considered to obtain a stable approximate solution. Error estimate is established for the regularised solution for different regularisation schemes. Numerical results that confirm the theoretical results are also presented.