Thomas Pascal Wihler

We study iterative finite element approximations for the numerical approximation of semilinear elliptic boundary value problems with monotone nonlinear reactions of subcritical growth. The focus of our contribution is on an optimal a priori error estimate for a contractive Picard type iteration scheme on meshes that are locally refined towards possible corner singularities in polygonal domains. Our analysis involves, in particular, an elliptic regularity result in weighted Sobolev spaces and the use of the Trudinger inequality, which is instrumental in dealing with subcritically growing nonlinearities. A series of numerical experiments confirm the accuracy and efficiency of our method.

We introduce a new hp -adaptive strategy for self-adjoint elliptic boundary value problems that does not rely on using classical a posteriori error estimators. Instead, our approach is based on a generally applicable prediction strategy for the reduction of the energy error that can be expressed in terms of local modifications of the degrees of freedom in the underlying discrete approximation space. The computations related to the proposed prediction strategy involve low-dimensional linear problems that are computationally inexpensive and highly parallelizable. The mathematical building blocks for this new concept are first developed on an abstract Hilbert space level, before they are employed within the specific context of hp -type finite element discretizations. For this particular framework, we discuss an explicit construction of p -enrichments and hp -refinements by means of an appropriate constraint coefficient technique that can be employed in any dimensions. The applicability and effectiveness of the resulting hp -adaptive strategy is illustrated with some 1- and 2-dimensional numerical examples.

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems, where the nonlinear reaction terms need to be neither monotone nor convex. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy reduction approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilized) linearized discrete problems, and tends to a critical point of the underlying energy functional in a stable way. Simultaneously, the finite-dimensional approximation spaces are adaptively refined. This is implemented in terms of a new mesh refinement strategy in the context of finite element discretizations, which again relies on the energy structure of the problem under consideration. In particular, in contrast to more traditional approaches, it does not involve any a posteriori error estimators, and is based on local energy reduction indicators instead. In combination, the resulting adaptive algorithm consists of an iterative linearization procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge toward a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.

The starting point of this note is a decades-old yet little-noticed sufficient condition, presented by Sassenfeld in 1951, for the convergence of the classical Gauss-Seidel method. The purpose of the present paper is to shed new light on Sassenfeld's criterion and to demonstrate that it is directly related to H-matrices. In particular, our results yield a new characterization of H-matrices. In addition, the convergence of iterative linear solvers that involve H-matrix preconditioners is briefly discussed.

The classical Kačanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Kačanov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilised) linearised discrete problems, and tends to a local minimum of the underlying energy functional. Simultaneously, the finite-dimensional approximation spaces are adaptively refined; this is implemented in terms of a new mesh refinement strategy in the context of finite element discretisations, which again relies on the energy structure of the problem under consideration, and does not involve any a posteriori error indicators. In combination, the resulting adaptive algorithm consists of an iterative linearisation procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge towards a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.

The classical Ka\v{c}anov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Ka\v{c}anov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 2020, 326, 2707–2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 2020, Paper No. 24] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the overall computational costs, preprint 2020]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.