Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method

Abstract

We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are secondorder accurate and first-order accurate, respectively, in the L ∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions.

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

“…In addition, a number of specialized methods have been designed to achieve better than first order accuracy in the L ∞ norm. These include the immersed interface (e.g., [20,21,22,23] [24,25]), ghost fluid (e.g., [26,27,28,29]), cut-cell methods (e.g., [30,31,32,33]) and Voronoi interface (e.g., [34,35]) methods. These methods modify difference stencils near the domain boundary to account for the boundary conditions.…”

Rational Krylov methods for functions of matrices with applications to fractional partial differential equations

“…Within the same framework, one can apply the proposed rational Krylov Jacobi method also to fractional linear multistep methods [37] for solving (2) in which the first order derivative in time is replaced by a fractional derivative of order β, with β ∈ (0, 1). 19 In principle, the proposed method can work on irregular domains, Robin or Dirichlet BC, see, e.g., [38][39][40], and can be used also in contexts of adaptivity, provided that these generate a symmetric positive definite matrix. In the latter case, studying how the selected poles vary as the mesh is changed would be of interest, and could also open new alternative approaches.…”

On two-phase flow solvers in irregular domains with contact line

“…This approach is widely used in the literature, even though it is only first-order accurate for solving the Poisson equation with jump conditions. In [21], a second-order approach developed, but it has not been applied to Navier-Stokes equations yet. The Ghost-Fluid Method introduced in [23] can be described briefly as proposed in [25]: first, the intermediate velocity u * 2 is updated with…”