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Error evaluations for controllable adaptive algorithms based on the Hessian recovery. (Russian, English) Zbl 1087.65113
Zh. Vychisl. Mat. Mat. Fiz. 45, No. 8, 1424-1434 (2005); translation in Comput. Math. Math. Phys. 45, No. 8, 1374-1384 (2005).
The goal is to analyze the interpolation errors for adaptive algorithms on the ground of automatically restored metrics. Let \(u \in C^0(\bar \Omega)\) and \(\mathcal P_{\Omega ^h}\) a certain projector in this space. A grid \(\Omega_{opt}^h(N_T, u)\) which consists of less than \(N_T\) tetrahedra is defined as an optimal one if it presents a solution to the optimization problem \[ \Omega_{opt}^h(N_T, u) = \arg_{\Omega_h : \mathcal N(\Omega_h) \leq N_T} \min | | u - \mathcal P_{\Omega ^h}u| | _{L_{\infty}(\Omega)}. \] A concept of quasi-optimal grid is introduced. A method of approximate solution to the optimization problem is based upon recovery of the Hessian of grid. The optimization problem is replaced by the mesh generation problem, which should be quasi-uniform in the metric space, which is generated by the Hessian of the \(\mathcal P_{\Omega ^h}u\) function.

MSC:
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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