Numerical solving the identification problem for the lower coefficient of parabolic equation

TL;DR

This paper develops a numerical method for identifying a time-dependent lower coefficient in a multidimensional parabolic PDE, using finite element approximations and elliptic problem solutions, demonstrated on a 2D model.

Contribution

It introduces a novel computational algorithm combining linearized time approximations and elliptic problem solutions for inverse coefficient identification.

Findings

Effective in solving inverse problems for 2D parabolic equations

Demonstrates accuracy of the method through numerical experiments

Provides a practical approach for multidimensional inverse PDE problems

Abstract

In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a nonlinear inverse problem, linearized approximations in time are constructed using standard finite element procedures in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems. The numerical results presented here for a model 2D problem demonstrate capabilities of the proposed computational algorithms for approximate solving inverse problems.