Isogeometric spectral approximation for elliptic differential operators
For researchers in computational mechanics and numerical analysis, this work provides an incremental improvement to isogeometric analysis by enhancing accuracy and robustness of spectral approximations.
This paper develops optimally blended quadrature rules for isogeometric spectral approximation of second-order elliptic eigenvalue problems, achieving two extra orders of super-convergence in eigenvalue error and minimizing dispersion error, as demonstrated in 1D and 3D numerical examples including the Schrödinger operator.
We study the spectral approximation of a second-order elliptic differential eigenvalue problem that arises from structural vibration problems using isogeometric analysis. In this paper, we generalize recent work in this direction. We present optimally blended quadrature rules for the isogeometric spectral approximation of a diffusion-reaction operator with both Dirichlet and Neumann boundary conditions. The blended rules improve the accuracy and the robustness of the isogeometric approximation. In particular, the optimal blending rules minimize the dispersion error and lead to two extra orders of super-convergence in the eigenvalue error. Various numerical examples (including the Schr$\ddot{\text{o}}$dinger operator for quantum mechanics) in one and three spatial dimensions demonstrate the performance of the blended rules.