Convergence Rate Analysis of Galerkin Approximation of Inverse Potential Problem

TL;DR

This paper provides a detailed convergence rate analysis for Galerkin finite element approximations of the inverse potential problem, including stability estimates, error bounds, and numerical validation.

Contribution

It introduces novel weighted stability estimates and comprehensive error analysis for Galerkin-based reconstruction of inverse potentials in elliptic and parabolic problems.

Findings

Weighted stability estimates under mild conditions

Explicit error bounds depending on noise and discretization

Numerical experiments confirming theoretical results

Abstract

In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an $H^1$-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed analysis of the discrete scheme, and provide convergence rates in a weighted $L^2(\Omega)$ for discrete approximations with respect to the exact potential. The error bounds are explicitly dependent on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard $L^2(\Omega)$ and interior $L^2$ norms. The analysis employs sharp a priori error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.