Efficient numerical methods for boundary data and right-hand side reconstructions in elliptic partial differential equations. (English) Zbl 1330.65167
Summary: 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. 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.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |
| 35R30 | Inverse problems for PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
dual reciprocity method; elliptic equation; inverse problem; method of fundamental solutions; regularization; Ritz; Galerkin method; numerical example; stabilityReferences:
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