Quasi-Newton variable preconditioning for nonlinear nonuniformly monotone elliptic problems posed in Banach spaces

Quasi-Newton-type iterative solvers are developed for a wide class of nonlinear elliptic problems. The presented generalization of earlier efficient methods covers various nonuniformly elliptic problems arising, e.g., in non-Newtonian flows or for certain glaciology models. The robust estimates are reinforced by several examples

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator

Fast computations with the harmonic Poincaré-Steklov operators on nested refined meshes

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincar'e-Steklov operators in presence of nested mesh refinement. For both interior and exterior problems the matrix-vector multiplication for the finite element approximations to the Poincar'e-Steklov operators is shown to have a complexity of the order O(Nreflog3N) where Nref is the number of degrees of freedom on the polygonal boundary under consideration and N = 2-p0 · Nref, p0 ≥ 1, is the dimension of a finest quasi-uniform level. The corresponding memory needs are estimated by O(Nreflog2N). The approach is based on the multilevel interface solver (as in the case of quasi-uniform meshes, see [20]) applied to the Schur complement reduction onto the nested refined interface associated with nonmatching decomposition of a polygon by rectangular substructures

Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems

This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp