A finite difference scheme with an optimal convergence for elliptic PDEs on domains defined by a level-set function
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In this talk, we will present a new finite difference method, on regular grid, well suited for elliptic problems posed in a domain given by a level-set function. It is inspired by the ϕ-FEM paradigm [1, 2, 3, 4] which is a fictitious finite element method imposing the boundary conditions thanks to a level-set function describing the domain. We will consider here the Poisson equation with Dirichlet condition. We will prove the optimal convergence of our finite difference method in some Sobolev norms. Moreover, the discrete problem is proven to be well conditioned, i.e. the condition number of the associated matrix is of the same order as that of a standard method on a comparable mesh. We will then give some numerical results that confirm the optimal convergence in the considered Sobolev norms. An other advantage of our approach is that it uses standard libraries such as Numpy and Scipy in Python, and the implementation is very short (less than 100 lines in Python), making it a very low-cost numerical scheme in terms of computation time.
