A fourth-order Cartesian grid embedded boundary method for Poisson’s equation

Abstract

In this paper, we present an fourth order algorithm to solve Poisson's equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second order algorithm. We also discuss in depth strategies for retaining higher order accuracy in the presence of non-smooth geometries.

Towards a High-Order Embedded Boundary Finite Volume Method for the Incompressible Navier-Stokes Equations with Complex Geometries

“…Vi While the EB method has been used in a number of complex fluid dynamics applications [12,2,22,8], typical applications have only achieved up to second-order accurate solutions, with first-order or inconsistent results near embedded boundaries [25]. There has been recent progress with improvements in accuracy [21], for example a high-order finite volume EB method for smooth and kinked (C 0 ) domains was demonstrated for Poisson's equation by Devendran et al [11]. The present work extends that fourth-order EB method to solve the time dependent Stokes equations.…”

A Finite Element Penalized Direct Forcing Immersed Boundary Method for infinitely thin obstacles in a dilatable flow

“…• Adds new terms: the Lagrange Multipliers method [13,3,14], the Penalty method [15][16][17], the Fat Boundary Method [18,19], the Immersed Spread Interface method [20,21], the Immersed Boundary Method [22][23][24], the diffuse domain approach [25], etc. • Modifies the operators: the Immersed Interface Method [26][27][28], the Ghost [29][30][31] and Cut Cell methods [32], the Cartesian Grid Embedded Boundary method [33][34][35], the Jump Embedded Boundary Conditions [36,21], the Finite Cell Method [37], the X-FEM-based FDMs [38], and so on.…”

An efficient Poisson solver for complex embedded boundary domains using the multi-grid and fast multipole methods

“…However, convergence (number of MG iterations) of their method depends on the number of unknowns, and the dependence is stronger for larger difference in coefficients across the EB. Devendran et al (2017) developed a fourth order EB Poisson solver based on the cut-cell approach and demonstrated convergence for sharp complex geometries (C 0 ) through geometric regularization. Algebraic multigrid method is a popular alternative for complex geometries, however, it suffers from significant memory/storage requirements.…”