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Center for Nonlinear Studies |
One of the main goals of unstructured mesh adaptation algorithms is to achieve better accuracy of a solution with a smaller number of mesh elements. For problems with anisotropic solutions, the optimal adaptive mesh must contain anisotropic elements. It has been shown in a number of papers that triangles with obtuse and acute angles stretched along the direction of minimal second derivative of a solution may be the best elements for minimizing an interpolation error. As the result, an optimal adaptive mesh may frequently contain anisotropic elements. The conventional approach to control anisotropic adaptivity is based on metric usage. One way to generate a metric is to use the Hessian (the matrix of second derivatives) recovered from a computed solution. It has been shown in [1,2] that the adaptive meshes quasi-uniform in the Hessian-based metric result in optimal estimates for the interpolation error. From practical point of view, the Hessian-based adaptation suggests a simple and user-friendly technology. A control of mesh adaptation is a natural component of the technique. The control is performed by a modification of the Hessian-based metric. New error estimates [3] give insight into effects of different metric modifications on the interpolation error. In the talk, I discuss briefly new theoretical results for simplicial (triangular in 2D and tetrahedral in 3D) meshes and present some numerical experiments showing technological appeal of the approach. 1. Y.Vassilevski, K.Lipnikov, Adaptive algorithm for generation of quasi-optimal meshes. Comp. Math. Math. Phys. 39, 1532-1551, 1999. 2. A. Agouzal, K. Lipnikov, Y. Vassilevski, Adaptive generation of quasi-optimal tetrahedral meshes. East-West J. Numer. Math., 7, 223-244, 1999. 3. Y.Vassilevski, K.Lipnikov, Error bounds for controllable adaptive algorithms based on a Hessian recovery. Comp. Math. Math. Phys. 45, 1374-1384, 2005.