A spectral embedding method applied to the advection-diffusion equation
Journal of Computational Physics
A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
Journal of Computational Physics
Short Note: The method of fundamental solutions for 2D and 3D Stokes problems
Journal of Computational Physics
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Spectral domain embedding for elliptic PDEs in complex domains
Journal of Computational and Applied Mathematics
Fictitious Domain approach with hp-finite element approximation for incompressible fluid flow
Journal of Computational Physics
A spectral fictitious domain method with internal forcing for solving elliptic PDEs
Journal of Computational Physics
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This work is a continuation of the authors efforts to develop high-order numerical methods for solving elliptic problems with complex boundaries using a fictitious domain approach. In a previous paper, a new method was proposed, based on the use of smooth forcing functions with identical shapes, mutually disjoint supports inside the fictitious domain and whose amplitudes play the role of Lagrange multipliers in relation to a discrete set of boundary constraints. For one-dimensional elliptic problems, this method shows spectral accuracy but its implementation in two dimensions seems to be limited to a fourth-order algebraic convergence rate. In this paper, a spectrally accurate formulation is presented for multi-dimensional applications. Instead of being specified locally, the forcing function is defined as a convolution of a mollifier (smooth bump function) and a Lagrange multiplier function (the amplitude of the bump). The multiplier function is then approximated by Fourier series. Using a Fourier Galerkin approximation, the spectral accuracy is demonstrated on a two-dimensional Laplacian problem and on a Stokes flow around a periodic array of cylinders. In the latter, the numerical solution achieves the same high-order accuracy as a Stokes eigenfunction expansion and is much more accurate than the solution obtained with a classical third order finite element approximation using the same number of degrees of freedom.