Detroit, MI 48202
Speaker: Sara Pollock, Wright State University
Title: Adaptive regularization for solution-dependent diffusion.
The approximation of solutions to elliptic solution-dependent diffusion problems presents unique challenges not encountered in linear problems, or in the related problems of nonlinear monotone type. The first main challenge is the instability of the iterative solution process. Ill-conditioned and potentially indefinite Jacobians may cause spurious modes to grow from one solution iterate to the next, leading to the divergence of direct linearization methods. The second main challenge is assuring the consistency of the discrete problem with respect to the continuous model. In particular, highly oscillatory layers in the diffusion coefficient may need to be uncovered during the solution process. We will discuss adaptive strategies based on incomplete solves of regularized problems, to solve the discrete nonlinear equations induced by finite element discretizations of these nonmonotone quasilinear PDE. Strategies for regularization parameter selection and exit criteria for the nonlinear iterations will be discussed in the context of running efficient simulations starting from a coarse mesh. Numerical examples will illustrate the ideas and demonstrate the presented adaptive algorithm.