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this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a second-order Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This newapproach is then applied to Hele-Shaw flow, an unstable flow involving two fluids with very different viscosity. 1997 Academic Press L

. A fast, second order accurate iterative method is proposed for the elliptic equation r \Delta (fi(x; y)ru) = f(x; y) in a rectangular region\Omega in 2 space dimensions. We assume that there is an irregular interface across which the coefficient fi, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the jump in fi is large. The interface may or may not align with a underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [SINUM, 4 (1994), pp. 1019-1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second order difference scheme for a corresponding Poisson equation in the region, and a second order discretization for a Ne...

New numerical methods using the finite element formulation with Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients and singularities in the source terms. The triangulations do not need to fit the interfaces. The idea is to construct basis functions which satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximal norm. Due to the special triangulations, the methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis. Second order convergence in the maximal norm is observed for the finite element method using the conforming finite element space.

The geometric construction of finite element spaces suitable for complicated shapes or microstructured materials is discussed. As an application, the efficient computation of linearized elasticity is considered on them. Geometries are supposed to be implicitly described via 3D voxel data (e. g. µCT scans) associated with a cubic grid. We place degrees of freedom only at the grid nodes and incorporate the complexity of the domain in the hierarchy of finite element basis functions, i. e. constructed by cut off operations at the reconstructed domain boundary. Thus, our method inherits the nestedness of uniform hexahedral grids while still being able to resolve complicated structures. In particular, the canonical coarse scales on hexahedral grid hierarchies can be used in multigrid methods. AMS Subject Classifications: 65N30, 65N55, 65N50. 1

We present a graph-based strategy for representing the computational domain for embedded boundary discretizations of conservation-law PDE's. The representation allows recursive generation of coarse-grid geometry representations suitable for multigrid and adaptive mesh re#nement calculations. Using this scheme, we implement a simple multigrid V-cycle relaxation algorithm to solve the linear elliptic equations arising from a block-structured adaptive discretization of the Poisson equation over an arbitrary two-dimensional domain. We demonstrate that the resulting solver is robust to a wide range of two-dimensional geometries, and performs as expected for multigrid-based schemes, exhibiting O #N log N# scaling with system size, N . Keywords: Cartesian grid, embedded boundary, adaptive mesh re#nement, multigrid, Poisson equation, linear solution methods 1 Introduction In the Embedded Boundary #EB# approach to discretizing PDE's in complex geometries, the physical domain is e...

I present a new algorithm for solving the streamfunction-vorticity equations in irregular, multiply connected regions. To avoid mesh generation difficulties associated with unstructured, body fitted grids, I embed the irregular domain in a uniform Cartesian mesh. The governing partial differential equations are discretized using standard finite volume and finite difference methods away from the irregular boundary. Near the irregular boundary, special discretizations are used to impose boundary conditions. I solve the vorticity transport equation using the high-resolution algorithms in the clawpack package and capacity form differencing to handle the irregular geometry. By modifying the capacity function for small cells cut by the boundary, I can avoid the small cell instability problem associ...

ves entering the cell. High-resolution extensions are defined by viewing these waves as a characteristic decomposition of the slope, and applying limiter functions to these waves before using them in a second-order correction term based on Taylor series expansion. By including a "transverse Riemann solver", a multi-dimensional algorithm is easily applied in 2 or 3 space dimensions which in some cases is more effective than dimensional splitting (which is also included as an option). The user must also provide a boundary condition routine that extends the solution to a set of ghost cells in each time step. A source term routine can also be provided, in which case a fractional step method is used to alternate between solving the homogeneous hyperbolic system and calling the source-term routine. Although it is well known that a "Strang splitting" should normally be used to maintain full second-order accuracy in fractional step methods, in the context of high-resolution methods for disco