\(C^0\mathrm{IPG}\) adaptive algorithms for the biharmonic eigenvalue problem
The \(C^0\mathrm{IPG}\) refers to \(C^0\) interior penalty Galerkin methods as initiated in [\textit{S. C. Brenner}, Lect. Notes Comput. Sci. Eng. 85, 79--147 (2012; Zbl 1248.65120)] and further elaborated in subsequent papers. Here the problem is to study the biharmonic eigenvalue problem \(\Delta^2 u=\lambda u\) in \(\Omega\), a bounded polygonal domain in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) while \(u\) and its outward normal derivative vanishes on the boundary \(\partial \Omega\). Using Lagrange finite elements on a shape-regular triangularization, Brenner [loc. cit.] also provides a posteriori error indicators for \(C^0\)IPG, which are proved in this paper to be reliable and efficient for the problem at hand. Moreover, the standard adaptive refinement algorithm, which shows an increase in the ratio \(h^{k-1}/h^\alpha_{\min}\) (\(h\) is the mesh size, \(k\) the degree of the method, and \(\alpha\in(0.5,1]\) the regularity index). This ratio should be \(O(1)\). By preventing an increase in this ratio above some threshold, a more uniform refinement algorithm is proposed. This is tested numerically on a unit square with a slit, and on an L-shaped domain.
\(C^0\) interior penalty Galerkin method
scientific article; zbMATH DE number 6894100
scientific article