A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems

“…Similar discretization approaches can be found in [32,33], in that all methods use Taylor series expansion for local approximation and use two physical jumps only. We note that [32,33] use a wider stencil in both 1D and higher dimensions and thus have trouble dealing with complex interfaces.…”

“…A fixed Cartesian grid, where the interface can cut through the grid lines, is often used. A variety of methods have been proposed to deal with the grid-interface interaction [4,7,9,10,16,[13][14][15]28,32,[29][30][31]33,34].…”

“…We note that [32,33] use a wider stencil in both 1D and higher dimensions and thus have trouble dealing with complex interfaces. For higher-dimensional cases, [32,33] uses dimension splitting approaches with one-sided interpolation and have to refine the mesh when multiple intersections occur. On the other hand, the PIM uses higher-dimensional polynomials for approximation and produces accurate results without refining the mesh for complex interfaces with multiple intersections.…”

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.…”

Numerical investigation of an oscillating airfoil using immersed boundary method

“…Immersed-boundary method, first developed by Peskin et al [13,14]., avoids re-gridding for moving boundary and thus becomes a promising scheme for complex and moving geometries. This method has been widely applied and developed in various research fields [15][16][17][18][19][20][21][22]. However, to author's knowledge, no work related to the oscillating airfoil simulation has been done.…”