Thomas P. Wihler

Mario Amrein, Thomas P. Wihler

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

Mario Amrein, Thomas P. Wihler

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.

Thomas P. Wihler, Bänz Bessire, André Stefanov

Given a large real symmetric, positive semidefinite m-by-m matrix, the goal of this paper is to show how a numerical approximation of the entropy, given by the sum of the entropies of the individual eigenvalues, can be computed in an efficient way. An application from quantum-optics illustrates the new algorithm.

Bärbel Holm, Thomas P. Wihler

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Mario Amrein, Jens M. Melenk, Thomas P. Wihler

In this paper we develop an $hp$-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an $hp$-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully $hp$-adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Lars Schmutz, Thomas P. Wihler

In this paper we investigate the $\mathrm{L}^\infty$-stability of fully discrete approximations of abstract linear parabolic partial differential equations. The method under consideration is based on an $hp$-type discontinuous Galerkin time stepping scheme in combination with general conforming Galerkin discretizations in space. Our main result shows that the global-in-time maximum norm of the discrete solution is bounded by the data of the PDE, with a constant that is robust with respect to the discretization parameters (in particular, it is uniformly bounded with respect to the local time steps and approximation orders).

Thomas P. Wihler

In this note we shall devise a variable-order continuous Galerkin time stepping method which is especially geared towards norm-preserving dynamical systems. In addition, we will provide an a posteriori estimate for the $L^\infty$-error.

Mario Amrein, Thomas P. Wihler

In this paper we investigate the application of pseudo-transient-continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, employ the PTC-methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction-type PTC-method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.

Thomas P. Wihler

Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that a suitable contraction property can be verified. However, such conditions are typically too restrictive for more complex nonlinear equations that lack key structural features such as monotonicity or convexity. In this paper, we develop a general framework for the weak convergence of fixed point iterations based on asymptotic bounds. In particular, we introduce and exploit a weak sequential non-expansiveness property, which is significantly weaker than the global Lipschitz assumptions commonly employed in this context. This approach permits to extend classical convergence results to a broader class of mappings in general (reflexive) Opial spaces, without relying on additional geometric assumptions such as uniform convexity.

Pascal Heid, Thomas P. Wihler

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in our previous work. The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws.

Ramona Baumann, Thomas P. Wihler

We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an $H^2$-regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order with respect to the mesh size.

Paul Houston, Thomas P. Wihler

In this paper we develop an $hp$-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an $hp$-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust $hp$-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Mario Amrein, Thomas P. Wihler

In this paper we develop an adaptive procedure for the numerical solution of semilinear parabolic problems, with possible singular perturbations. Our approach combines a linearization technique using Newton's method with an adaptive discretization-which is based on a spatial finite element method and the backward Euler time stepping scheme-of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Paul Houston, Thomas P. Wihler

In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this way we aim to account for local smoothness properties of the function to be integrated as effectively as possible, and thereby achieve highly accurate results in a very efficient manner. Indeed, this idea originates from so-called hp-version finite element methods which are known to deliver high-order convergence rates, even for nonsmooth functions.

Paul Houston, Thomas P. Wihler

This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of $hp$-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new $hp$-refinement technique for both one- and two-dimensional problems.

Scott Congreve, Thomas P. Wihler

In this article we investigate a finite element formulation of strongly monotone quasi-linear elliptic PDEs in the context of fixed-point iterations. As opposed to Newton's method, which requires information from the previous iteration in order to linearise the iteration matrix (and thereby to recompute it) in each step, the alternative method used in this article exploits the monotonicity properties of the problem, and only needs the iteration matrix calculated once for all iterations of the fixed-point method. We outline the a priori and a posteriori error estimates for iteratively obtained solutions, and show both theoretically as well as numerically how the number of iterations of the fixed-point method can be restricted in dependence of the mesh size, or of the polynomial degree, to obtain optimal convergence.

Jens M. Melenk, Thomas P. Wihler

We consider the approximation of singularly perturbed linear second-order boundary value problems by $hp$-finite element methods. In particular, we include the case where the associated differential operator may not be coercive. Within this setting we derive an a posteriori error estimate for a natural residual norm. The error bound is robust with respect to the perturbation parameter and fully explicit with respect to both the local mesh size $h$ and the polynomial degree $p$.