Direct method of solution for general boundary value problem of the Laplace equation

A first order method for the Cauchy problem for the Laplace equation using BEM

“…The boundary mesh is a regular one with 120 elements on each side. Comparison with other methods This example was already studied by our previous method [6], a Tikhonov like method [11] and an iterative method [16] which all gave similar results. Figure 15 (respectively Figure 16) compares function (respectively normal derivatives) reconstructions obtained with one of these methods [6] and with the first order method.…”

“…Other methods exist to solve this problem. The main methods are Tikhonov like methods, [8], [11], [17], [18], quasi reversibility methods [3], [9], [13], [15], and iterative methods [1], [12], [14], [16]. Quasi reversibility methods and Tikhonov regularization methods present the avantage to lead to well-posed problems after modifying the partial derivative operator.…”

A robust data completion method for 2D Laplacian Cauchy problems

“…In the first case, the data are not noisy and are prescribed on half of the boundary. It was studied in [13] using a Tikhonov like method , in [4] using a fading regularization method and in [8] using an iterative method. All these methods gave similar results.…”

The PCGM for Cauchy inverse problems in 3D potential

“…The influence of measurement location, measurement error and element option were investigated. An inverse problem for Laplace equations was recast into primary and adjoint boundary value problems by Hayashi et al [8,9]. The Dirichlet and Neumann data were specified on respective part of the boundary, while no data on the second part of the boundary were given and Robin condition was prescribed on the third part of the boundary.…”