Ryan Szypowski

Michael Holst, Ryan Szypowski, Yunrong Zhu

In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such as the nonlinear Poisson-Boltzmann equation and its regularizations. The algorithm we study consists of the standard SOLVE-ESTIMATE-MARK-REFINE procedure common to many adaptive finite element algorithms, but where the SOLVE step involves only a full solve on the coarsest level, and the remaining levels involve only single Newton updates to the previous approximate solution. We summarize a recently developed AFEM convergence theory for inexact solvers, and present a sequence of numerical experiments that give evidence that the theory does in fact predict the contraction properties of AFEM with inexact solvers. The various routines used are all designed to maintain a linear-time computational complexity.

Michael Holst, Ryan Szypowski, Yunrong Zhu

In this article we consider two-grid finite element methods for solving semilinear interface problems in d space dimensions, for d=2 or d=3. We first describe in some detail the target problem class with discontinuous diffusion coefficients, which includes problems containing sub-critical, critical, and supercritical nonlinearities. We then establish basic quasi-optimal a priori error estimate for Galerkin approximations. In the critical and subcritical cases, we follow our recent approach to controling the nonlinearity using only pointwise control of the continuous solution and a local Lipschitz property, rather than through pointwise control of the discrete solution; this eliminates the requirement that the discrete solution satisfy a discrete form of the maximum principle, hence eliminating the need for restrictive angle conditions in the underlying mesh. The supercritical case continues to require such mesh conditions in order to control the nonlinearity. We then design a two-grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm, and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. The algorithm we describe, and its analysis in this article, combines four sets of tools: the work of Xu and Zhou on two-grid algorithms for semilinear problems; the recent results for linear interface problems due to Li, Melenk, Wohlmuth, and Zou; recent work on the Poisson-Boltzmann equation; and recent work on a priori estimates for semilinear problems.

Randolph E. Bank, Michael Holst, Ryan Szypowski et al.

In this article we consider a priori error and pointwise estimates for finite element approximations of solutions to semilinear elliptic boundary value problems in d>=2 space dimensions, with nonlinearities satisfying critical growth conditions. It is well-understood how mesh geometry impacts finite element interpolant quality, and leads to the reasonable notion of shape regular simplex meshes. It is also well-known how to perform both mesh generation and simplex subdivision, in arbitrary space dimension, so as to guarantee the entire hierarchy of nested simplex meshes produced through subdivision continue to satisfy shape regularity. However, much more restrictive angle conditions are needed for basic a priori quasi-optimal error estimates, as well as for a priori pointwise estimates. These angle conditions, which are particularly difficult to satisfy in three dimensions in any type of unstructured or adaptive setting, are needed to gain pointwise control of the nonlinearity through discrete maximum principles. This represents a major gap in finite element approximation theory for nonlinear problems on unstructured meshes, and in particular for adaptive methods. In this article, we close this gap in the case of semilinear problems with critical or sub-critical nonlinear growth, by deriving a priori estimates directly, without requiring the discrete maximum principle, and hence eliminating the need for restrictive angle conditions. Our main result is a type of local Lipschitz property that relies only on the continuous maximum principle, together with the growth condition. We also show that under some additional smoothness assumptions, the a priori error estimate itself is enough to give pointwise control the discrete solution, without the need for restrictive angle conditions. Numerical experiments confirm our theoretical conclusions.

Michael Holst, Yuwen Li, Adam Mihalik et al.

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in two dimensions, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.