A spectral fictitious domain method with internal forcing for solving elliptic PDEs

On the spectral accuracy of a fictitious domain method for elliptic operators in multi-dimensions

“…Unfortunately, in that case, the solution of (2) is no longer smooth across c and so spectral methods lose their property of exponential convergence. In a previous paper, [5] have proposed a new formulation for f. The idea of this method (FDMIF method) is to replace each delta function at the boundary by a smooth function h q defined on a ball of some radius q inside the fictitious domain. The centers of the balls are taken on some ðn À 1Þ dimensional manifoldc contained in x.…”

“…The function k, defined onc, is the Lagrange multiplier function associated with the constraint u jc ¼ u 0 on c. The smooth function used by [5] is the infinitely differentiable bump function (also called ''mollifier'') whose support is a ball of radius q:…”

“…Buffat and Le Penven [5] have applied the FDMIF method, with a particular choice for (3) in which k is expressed as a sum of Dirac masses onc. For the different problems investigated, the calculated solution is smooth across the fictitious boundary and a high-order accuracy is recovered.…”

“…In a previous paper, Buffat and Le Penven [5] proposed a new fictitious domain method with internal forcing inside the fictitious domain (FDMIF), allowing the accuracy of spectral methods to be preserved, unlike the classical discrete Lagrange multiplier (DLM) method [9,10]. By solving one-dimensional test problems for second-and fourth-order elliptic PDEs, they demonstrated the spectral convergence rate of the FDMIF.…”

“…In Section 2, we describe the basic idea of the method in the context of second-order elliptic problems. In Section 3, we implement the method in two dimensions using a spectral Fourier approximation instead of the Fourier-Chebyshev approximation used in [5]. Different approximation methods for the Lagrange multiplier function and the boundary constraint are presented in Section 4.…”