Adaptive FEM for parameter-errors in elliptic linear-quadratic parameter estimation problems

We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a posteriori error estimator is presented. Unlike prior results in the literature, our estimator, which is composed of standard energy error residual estimators for the state equation and suitable co-state problems, reflects the faster convergence of the parameter error compared to the (co)-state variables. We show optimal convergence rates of our method; in particular and unlike prior works, we prove that the estimator decreases with a rate that is the sum of the best approximation rates of the state and co-state variables. Experiments confirm that our method matches the convergence rate of the parameter error

hphp-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree $p \ge 1$ and the (local) mesh size $h$. We further prove that the built-in algebraic error estimator which comes with the solver is $hp$-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings

Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost

Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs

We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh refinement as well as the iterative linearization of the arising nonlinear discrete equations. To this end, we employ a damped Zarantonello iteration so that, in each step of the algorithm, only a linear Poisson-type equation has to be solved. We prove that the proposed AILFEM strategy guarantees convergence with optimal rates, where rates are understood with respect to the overall computational complexity (i.e., the computational time). Moreover, we formulate and test an adaptive algorithm where also the damping parameter of the Zarantonello iteration is adaptively adjusted. Numerical experiments underline the theoretical findings

On instance optimality of adaptive 2D FEM

Abweichender Titel nach Übersetzung der Verfasserin/des VerfassersZiel dieser Arbeit ist der Beweis der Instanzoptimalität adaptiver Finite Elemente Methoden (AFEM) für verschiedene Modellprobleme. Aufbauend auf dem Begriff der Populationen, der es erlaubt, bestimmte geometrische Eigenschaften von Gittern und deren Knotenmengen zu beweisen, sowie dem Begriff der Energie, die eng mit der Finite Elemente Lösung auf einem Gitter und deren Approximationsfehler verbunden ist, wird ein abstraktes Framework geschaffen, um Instanzoptimalität einer AFEM zu zeigen. Drei Eigenschaften werden sich hier als hinreichend erweisen: eine Lower Diamond Estimate der Energie, diskrete lokale Äquivalenz von Energie und a posteriori Fehlerschätzer, sowie eine Forderung an den Markierungsschritt. Diese Eigenschaften werden für zwei Modellprobleme nachgewiesen. Als Erstes werden elliptische Diffusionsprobleme mit gemischten Neumannund homogenen Dirichlet-Randbedingungen betrachtet, die durch konforme Finite Elemente beliebiger Ordnung diskretisiert werden. Weiter wird das abstrakte Framework auf zielorientierte adaptive Finite Elemente Methoden (GOAFEM) angewendet, bei denen die interessierende Größe der Funktionalwert eines linearen Funktionals der Lösung ist. Um den Fehler dieses Funktionalwerts abzuschätzen, wird eine modifizierte Fehlergröße eingeführt und für diese Instanzoptimalität gezeigt. Abschließend werden die Resultate einiger numerischer Experimente angegeben. Insgesamt verallgemeinert die Diplomarbeit die Arbeit [Diening, Kreuzer, Stevenson; Found. Comput. Math. 16 (2016)], in der Instanzoptimalität für eine adaptive P1-FEM für das Poisson-Problem mit homogenen Dirichlet Randbedingungen gezeigt wird.This thesis aims to prove instance optimality of adaptive finite element methods (AFEMs) for various model problems. Based on the concept of populations, which allows for the proof of certain geometric properties of meshes and their sets of nodes, as well as the concept of energy, which is closely related to the finite element solution of a mesh and its approximation error, an abstract framework is developed for proving instance optimality of an AFEM for selected problems. Three properties will turn out to be sufficient for instance optimality: a lower diamond estimate of the energy, discrete local equivalence of energy and a posteriori error estimator, as well as an assumption on the marking step. These properties will be shown to be valid for two model problems. First, elliptic diffusion problems with mixed Neumann and homogeneous Dirichlet boundary conditions will be considered, which are discretised by conforming finite elements of arbitrary order. Furthermore, the abstract framework will be applied to goal oriented adaptive finite element methods (GOAFEM), in which the quantity of interest is the value of a linear functional of the solution. To estimate the error of this value, a modified error quantity is introduced, for which instance optimality is shown. Finally, the theoretical findings are underpinned by numerical experiments. Overall, the present diploma thesis generalizes the work [Diening, Kreuzer, Stevenson; Found. Comput. Math. 16 (2016)], which proves instance optimality of an adaptive P1-FEM for the Poisson problem with homogeneous Dirichlet boundary conditions.9

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

We consider a general nonsymmetric second-order linear elliptic PDE in the framework of the Lax-Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive mesh-refinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method (AISFEM) leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory