**Eldorado - Repository of the TU Dortmund**

Resources for and from Research, Teaching and Studying

### Recent Submissions

It is an open question if the threshold condition θ < θ_* for the Dörﬂer marking parameter is necessary to obtain optimal algebraic rates of adaptive ﬁnite element methods. We present a (non-PDE) example ﬁtting into the common abstract convergence framework (axioms of adaptivity) and which is potentially converging with exponential rates. However, for Dörfler marking θ > θ_* the algebraic converges rate can be made arbitrarily small.

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is optimize the displacement ﬁeld in the domain occupied by the body by means of prescribed Dirichlet boundary data, which serve as control variables. The arising optimization problem is nonsmooth for several reasons, in particular, since the control-to-state mapping is not single-valued. We therefore apply a Yosida regularization to obtain a ...

This paper is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. So-called strong stationarity conditions are derived in an abstract framework. Strong stationarity conditions are regarded as the most rigorous ones, since they imply all other types of stationarity concepts and are equivalent to purely primal optimality conditions. The abstract framework is afterwards applied to four application-driven exa...

We approximate the solution of the Stokes equations by a new quasi-optimal and pressure robust discontinuous Galerkin discretization of arbitrary order. This means quasi-optimality of the velocity error independent of the pressure. Moreover, the discretization is well-defined for any load which is admissible for the continuous problem and it also provides classical quasioptimal estimates for the sum of velocity and pressure errors. The key design principle is a careful discretization of the l...

In this work, we introduce algebraic ﬂux correction schemes for enriched (P1 ⊕ P0 and Q1 ⊕ P0) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P1/Q1 component in the admissibl...

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress ﬁeld by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulﬁlled so that the system admits a solution, whose stress ﬁeld is unique. This gives rise to a well deﬁned control-to-...

To benefit from current trends in HPC hardware, such as increasing avail-ability of low precision hardware, we present the concept of prehandling as a direct way of preconditioning and the hierarchical finite element method which is exceptionally well-suited to apply prehandling to Poisson-like problems, at least in 1D and 2D. Such problems are known to cause ill-conditioned stiffness matrices and therefore high computational errors due to round-off. We show by means of numerical results that...

This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order ﬁnite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipula-tions of element contributions to the global nonlinear system. The required modiﬁcations can be carried out without calculating the element matrices and assembling their global counterparts. The co...

We present a time-simultaneous multigrid scheme for parabolic equations that is motivated by blocking multiple time steps together. The resulting method is closely related to multigrid waveform relaxation and is robust with respect to the spatial and temporal grid size and the number of simultaneously computed time steps. We give an intuitive understanding of the convergence behavior and briefly discuss how the theory for multigrid waveform relaxation can be applied in some special cases. Fin...

In this paper we present a holistic software approach based on the FEAT3 software for solving multidimensional PDEs with the Finite Element Method that is built for a maximum of performance, scalability, maintainability and extensibilty. We introduce basic paradigms how modern computational hardware architectures such as GPUs are exploited in a numerically scalable fashion. We show, how the framework is extended to make even the most recent advances on the hardware market accessible to the fr...

In this work, the novel “Tensor Diffusion” approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized “Tensor Stokes” problem, avoiding the need of considering a separate stress tensor in ...

In this work, the novel "Tensor Diffusion" approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized "Tensor Stokes" problem, avoiding the need of considering a separate stress tensor in ...

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...