A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources

By Benjamin L. Vaughan, Bryan G. Smith, David L. Chopp. 2006

1,392 citations recorded by the source corpus.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/01ecbbc9bf1a1be6e0253cca01fde9e73523f5a3

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In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes (2012, methodology): “The more recent Immersed Interface Method achieves higher order accuracy by avoiding the use of the Dirac delta distribution to define the forcing terms (see [22, 23, 31]).”
Three-dimensional elliptic solvers for interface problems and applications (2003, methodology): “We refer the readers to [7, 8, 9] for more discussions on the jump conditions.”
An immersed boundary technique for simulating complex flows with rigid boundary (2007, methodology): “This method was originally developed by LeVeque and Li [9] for solving elliptic equations with discontinuous coefficients and singular sources along an arbitrary interface.”
A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations (2009, methodology): “In practice, the most commonly used discrete delta functions are the 2-point hat function [20], the 4-point cosine function [2], the 3-point discrete delta function [21], and the 4-point piecewise function [2].”
A survey on level set methods for inverse problems and optimal design (2005, neutral): “We refer elsewhere [78, 88, 89, 119] for further details.”
Curvilinear Immersed Boundary Method for Simulating Fluid Structure Interaction with Complex 3D Rigid Bodies (2008, methodology): “In these methods the immersed boundary is treated as a sharp interface and its effect on the fluid is accounted for either by locally modifying the shapes of the grid cells to produce a boundary-fitted mesh (cut-cell methods [20]), or by incorporating the jump conditions caused by the discrete delta”
A combined extended finite element and level set method for biofilm growth (2008, background): “A number of finite difference based approaches for solving elliptic equations in conjunction with moving interfaces have been proposed, including the immersed interface method [21], the immersed boundary method [22], and the ghost fluid method [23].”
An augmented approach for Stokes equations with a discontinuous viscosity and singular forces (2007, methodology): “idea is to introduce two augmented variables that are defined only along the interface so that the jump conditions can be decoupled and the immersed interface method [11, 13] can be applied.”
Variational approach to shape derivatives for a class of Bernoulli problems (2006, background): “Leveque [9,10] on a rectangula with mesh sizeh = 2 49.”
A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem (2005, methodology): “Finally, we note that [21] designed a method that also yields a nonsymmetric linear system, but which is only second order accurate.”
Approximation capability of a bilinear immersed finite element space (2008, methodology): “Since then, FD methods such as the Cartesian grid method, embedded boundary method, cut-cell methods, etc., have been developed, and they have been used to treat Euler ∞ows [3, 25, 28, 42, 56], Navier-Stokes ∞ow [20, 23, 43], and of course, the interface problems involving the fundamental elliptic o”
IMMERSED BOUNDARY METHODS (2005, methodology): “General Considerations The methods presented in this section and other related methods not discussed here (Fedkiw & Liu 2000, Leveque & Li 1994) introduce the boundary condition directly into the discrete equations.”