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**A moving mesh finite element algorithm for singular problems in two and three space dimensions.**
*(English)*
Zbl 0998.65105

Summary: A framework for adaptive meshes based on the Hamilton-Schoen-Yau theory was proposed by A. S. Dvinsky [ibid. 95, No. 2, 450-476 (1991; Zbl 0733.65074)]. In a recent work [ibid. 170, No. 2, 562-588 (2001; Zbl 0986.65090)] we extended Dvinsky’s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability. In this work, we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions. In obtaining the variational mesh, we will solve an optimization problem with some appropriate constraints, which is in contrast to the traditional method of solving the Euler-Lagrange equation directly.

The key idea of this approach is to update the interior and boundary grids simultaneously, rather than considering them separately. Application of the proposed moving mesh scheme is illustrated with some two- and three-dimensional problems with large solution gradients. The numerical experiments show that our methods can accurately resolve detail features of singular problems in 3D.

The key idea of this approach is to update the interior and boundary grids simultaneously, rather than considering them separately. Application of the proposed moving mesh scheme is illustrated with some two- and three-dimensional problems with large solution gradients. The numerical experiments show that our methods can accurately resolve detail features of singular problems in 3D.

### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

### Keywords:

finite element method; moving mesh method; harmonic map; optimization; Hamilton-Schoen-Yau theory; mesh adaptation
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\textit{R. Li} et al., J. Comput. Phys. 177, No. 2, 365--393 (2002; Zbl 0998.65105)

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