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The paper is concerned with an optimal control problem governed by a state equa-tion in form of a generalized abstract operator differential equation involving a maximal monotoneoperator. The state equation is uniquely solvable, but the associated solution operator is in generalnot Gˆateaux-differentiable. In order to derive optimality conditions, we therefore regularize the stateequation and its solution operator, respectively, by means of a (smoothed) Yosida approximation.We show conver...
We investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patches Ω_k with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces X_h,k on each patch Ω_k as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of [12] and highlights several aspects which did not receive full attention be...
New results on the theoretical solvability of nonlinear algebraic flux correction (AFC) problems are presented and a Newton-like solution technique exploiting an efficient computation of the Jacobian is introduced. The AFC methodology is a rather new and unconventional approach to algebraically stabilize finite element discretizations of convection-dominated transport problems in a bound-preserving manner. Besides investigations concerning the theoretical solvability, the development of effic...
The aim of this paper is to describe a new, fast and robust solver for 3D flow problems which are described by the incompressible Navier-Stokes equations. The correspondig simulations are done by a monolithic 3D flow solver, i.e. velocity and pressure are solved at the same time. During these simulations the convective part is linearized using two different methods: Fixpoint method and Newton method. The Fixpoint method is working in a quite robust way, but it has a slow convergence dependi...
This paper is concerned with a priori error estimates for the local incremental minimization scheme, which is an implicit time discretization method for the approximation of rate-independent systems with non-convex energies. We first show by means of a counterexample that one cannot expect global convergence of the scheme without any further assumptions on the energy. For the class of uniformly convex energies, we derive error estimates of optimal order, provided that the Lipschitz const...
Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations, nonlinear scalar conservation laws, and hyperbolic systems. In contrast to fluxcorrected transport (FCT) algorithms that apply limited antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions...
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regulariza...
We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and sho...
In this work, we introduce a new residual distribution (RD) framework for the design of matrix-free bound-preserving finite element schemes. As a starting point, we consider continuous and discontinuous Galerkin discretizations of the linear advection equation. To construct the corresponding local extremum diminishing (LED) approximation, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high...
This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in [EM06], which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions f...
In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the following two properties. First, it is dominated by the error, irrespective of mesh fineness and the regularity of data and the exact solution. Second, it captures in terms of data the part of the residual that, in general, cannot be quantified with finite...
The article at hand focuses on finite element discretizations, where the continuous and the discrete formulations differ. We introduce a general approach based on the dual weighted residual method for estimating on the one hand the discretization error in a user specified quantity of interest and on the other hand the discrete model error induced by using different discrete techniques. Here, the usual error identities are obtained plus some additional terms. Furthermore, the numerical approxi...