Spectral stochastic finite elements applied to a few nonlinear problems
We study applications of spectral stochastic finite element methods (SSFEM) to eigenvalue problems and the Navier-Stokes equations. In the first part we focus on random eigenvalue problems. Given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration and Newton iteration, which are based on the stochastic Galerkin finite element method. Our approach allows the computation of interior eigenvalues, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. In the second part we focus on the steady-state Navier-Stokes equations assuming that the viscosity is a random field given by a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration. For both applications we compare the accuracy with that of Monte Carlo and stochastic collocation methods, and we demonstrate effectiveness of the algorithms using a set of benchmark problems. We will also remark on recent extensions to nonsymmetric eigenvalue problems and time-dependent Navier-Stokes equations. This is a joint work with Howard C. Elman, Kookjin Lee and Randy Price. The research was supported by the U.S. National Science Foundation under grant DMS1913201.
Math, 402 and Zoom https://arizona.zoom.us/j/85014462076
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