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High resolution sharp computational methods for elliptic and parabolic problems in complex geometries. (English) Zbl 1263.65093

Summary: We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.

MSC:

65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
80A22 Stefan problems, phase changes, etc.
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35R35 Free boundary problems for PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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References:

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