This work was partially supported by the AFOSR (through AFRL Cooperative Agreement #18RD- COR018, under grant FA9451-18-2-0031), the Croatian Science Foundation grant HRZZ-9345, bilateral Croatian-USA grant (administered jointly by Croatian-MZO and NSF) and NSF grant DMS-1522471. The numerical studies were facilitated by the equipment acquired using NSF’s Major Research Instrumentation grant DMS-1624776.
Eigenvalues -- Data processing, Galerkin methods, Discretization (Mathematics), Finite element method
A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.
Gopalakrishnan, J., Grubišić, L., Ovall, J., & Parker, B. Q. (2019). Analysis of FEAST spectral approximations using the DPG discretization. arXiv preprint arXiv:1901.07724.