A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Abstract

Keywords: Elliptic interface problems Embedded interface methods Virtual node methods Variational methods Multigrid methods a b s t r a c tWe present a numerical method for the variable coefficient Poisson equation in threedimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference Poisson stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L 1 -norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. [1] and introduces innovations necessary in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for cell-local polyhedral approximation to the zero isocontour surface of a level set needed for three-dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in [1], this is used in a discontinuity removal technique that yields the standard 7-point stencil across the interface and only requires a modification to the right-hand side of the linear system.

Analysis of the diffuse-domain method for solving PDEs in complex geometries

“…However, most of these methods typically require tools not frequently available in standard finite element and finite difference software packages. Examples of such approaches include the extended and composite finite element methods (e.g., [31,12,23,13,32,55,7,4]), immersed interface methods (e.g., [40,43,60,44,65]), virtual node methods with embedded boundary conditions (e.g., [3,73,34]), matched interface and boundary methods (e.g., [71,68,69,67,72]), modified finite volume/embedded boundary/cut-cell methods/ghost-fluid methods (e.g., [27,36,19,25,26,35,47,70,48,37,46,64,49,9,10,52,53,33,63]). In another approach, known as the fictitious domain method (e.g., [28,29,56,45]), the original system is either augmented with equations for Lagrange multipliers to enforce the boundary conditions, or the penalty method is used to enforce the boundary condi-tions weakly.…”

Preserving geometry and topology for fluid flows with thin obstacles and narrow gaps

“…IIM can achieve higher accuracy near boundaries when compared with IBM because it is able to successfully reduce the effects of the transfer function smearing due smooth kernels. The IIM is one of the most popular second-order finite difference methods for approximating interface problems (HELLRUNG et al, 2012).…”

“…To extend this strategy to multiple distinct flow regions within a single regular grid cell, as produced by our mesh clipping strategy, we take inspiration from recent virtual node (MOLINO; BAO;HELLRUNG et al, 2012) andtopology-preserving (TERAN et al, 2005;NESME et al, 2009) schemes. We allow multiple disjoint active sub-cells within a single original cell, with additional pressure and velocity degrees of freedom that conceptually coincide for consistency with Ng's discretization (see Figure 3.4).…”

A level set method for ductile fracture

“…We evaluate these integrands analytically over the polyhedral subelement embedded material regions. This is done with the divergence theorem as in [Hellrung et al 2012].…”