This work aims to investigate the application of the boundary element method (BEM), particularly its direct integration formulation (DIBEM), in solving inverse problems governed by the Laplace equation. Since the accuracy of inverse solutions obtained via the Monte Carlo method with Markov Chains (MCMC) strongly depends on the precision of the direct numerical solution, the study seeks to assess how BEM can enhance the quality of estimated variables in systems related to the diffusion equation, such as heat conduction, mass transport and groundwater contamination.
The study explores inverse problems solved using the MCMC and employs BEM as the direct numerical solver. Emphasis is placed on the DIBEM, which converts all domain terms into boundary terms and accurately evaluates integral representations without additional approximations. By applying this framework to Laplace equation-governed systems, the methodology examines the impact of BEM's numerical precision on the reliability of the inverse solution.
The analysis indicates that the precision of the BEM, especially when using DIBEM, can significantly improve the accuracy of inverse problem solutions obtained through MCMC. Because BEM eliminates the need for auxiliary domain-discretization methods and provides highly accurate boundary-based formulations, it enhances the estimation of unknown variables in problems governed by the Laplace equation. For the problem studied, numerical experiments yielded errors smaller than 1%, demonstrating the method's robustness. These characteristics make BEM a reliable tool for inverse analysis in applications such as heat conduction, electric and magnetic fields, and porous media flow.
The study highlights the combined use of the BEM and the MCMC in addressing inverse problems governed by the Laplace equation. While BEM is well established for direct analysis, its integration with MCMC for inverse formulations receives less attention in the literature. By emphasizing the advantages of DIBEM – particularly the ability to treat domain effects without approximations – the work provides a novel perspective on improving accuracy and reliability in inverse modeling across multiple physical applications.
