Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Abstract

In this article, we discuss the application of two important numerical methods, Ritz-Galerkin and Method of Fundamental Solutions (MFS), for solving some inverse problems, arising in the context of two-dimensional elliptic equations. The main incentive for studying the considered problems is their wide applications in engineering fields. In the previous literature, the use of these methods, particularly MFS for right hand side reconstruction has been limited, partly due to stability concerns. We demonstrate that these diculties may be surmounted if the aforementioned methods are combined with techniques such as dual reciprocity method(DRM). Moreover, we incorporate some iterative regularization techniques. This fact is especially veried by taking into account the noisy data with boundary conditions. In addition, parts of this article are dedicated to the problem of boundary data approximation and the issue of numerical stability, ending with a general discussion on the advantages and disadvantages of various methods.

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

“…To contribute the boundary conditions (1.3) in computations, we define the satisfier function 54,55,59,66,79 as s(x, t) = g 1 (t) + x(g 2 (t) − g 1 (t)) and consider the approximate solution of u(x, t) as…”

“…Operational matrices have been used as an efficient tool in the context of spectral methods such as Tau method and Ritz-Galerkin method. 36,37,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] These techniques provide a general approach based on expanding the approximate solution of the unknown functions in terms of the basis functions with unknown coefficients. Then, the operational matrices of the integration, differentiation, and product regarding the considered basis functions are utilized to tackle integral, differential, and nonlinear terms included in the problem.…”

“…To contribute the boundary conditions () in computations, we define the satisfier function 54,55,59,66,79 as s ( x , t ) = g 1 ( t ) + x ( g 2 ( t ) − g 1 ( t ))and consider the approximate solution of u ( x , t )as where and are the vectors including the orthonormal basis functions defined over the intervals [0, 1]and [0, t f ], respectively and is the coefficient matrix including the unknown parameter...…”

“…Operational matrices have been used as an efficient tool in the context of spectral methods such as Tau method and Ritz–Galerkin method 36,37,44–62 . These techniques provide a general approach based on expanding the approximate solution of the unknown functions in terms of the basis functions with unknown coefficients.…”