arXiv subjects

Olaf Steinbach

Publications and source records attributed to Olaf Steinbach.

At least 19 recordsLinked to original sources

arXiv2026

Space-time tensor-product finite element methods for parabolic problems

We study space-time Galerkin--Petrov formulations for parabolic evolution problems and their relation to classical implicit time-stepping schemes. Although such schemes are stable in the usual time-stepping sense, their interpretation as space-time operator equations may lead to conditional stability, with constants depending on the relation between temporal and spatial mesh sizes. We revisit this phenomenon for the continuous Galerkin method of Aziz and Monk, which yields the Crank--Nicolson scheme in the lowest-order case, and provide a detailed space-time error analysis for solutions of both high and low regularity. In particular, the space-time framework allows us to analyze the deteriorated behaviour of classical time-stepping methods for nonsmooth initial data. By applying integration by parts in time, we derive an adjoint space-time formulation that incorporates the initial condition in a natural variational way. In the lowest-order case, this formulation leads to a Rannacher-type smoothing of the initial data. The theoretical results are complemented by numerical experiments.

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arXiv2025

Numerical solution of elliptic distributed optimal control problems with boundary value tracking

We consider some boundary value tracking optimal control problem constrained by a Neumann boundary value problem for some elliptic partial differential equation where the control acts as right-hand side. This optimal control problem can be reformulated asa state-based variational problem that is the starting point for the finite element discretizion. In this paper, we only consider atensor-product finite element discretizion for which optimal discretization error estimates and fast solvers can be derived.Numerical experiments illustrate the theoretical results quantitatively.

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arXiv2025

Adaptive least-squares space-time finite element methods for convection-diffusion problems

In this paper we formulate and analyse adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative $\mathbf{v} \cdot \nabla u$ is considered as part of the total time derivative $\frac{d}{dt}u = \partial_t u + \mathbf{v} \cdot \nabla u$, and therefore we can use a rather standard stability and error analysis for related space-time finite element methods. For stationary problems we restrict the ansatz space $H^1_0(\Omega)$ such that the convective derivative is considered as an element of the dual $H^{-1}(\Omega)$ of the test space $H^1_0(\Omega)$, which also allows unbounded velocities $\mathbf{v}$. While the discrete finite element schemes are always unique solvable, the numerical solutions may suffer from a bad approximation property of the finite element space when considering convection dominated problems, i.e., small diffusion coefficients. Instead of adding suitable stabilization terms, we aim to resolve the solutions by using adaptive (space-time) finite element methods. For this we introduce a least-squares approach where the discrete adjoint defines local a posteriori error indicators to drive an adaptive scheme. Numerical examples illustrate the theoretical considerations.

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arXiv2025

A unified framework for the analysis, numerical approximation and model reduction of linear operator equations, Part I: Well-posedness in space and time

We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach towards numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.

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arXiv2025

State-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation

We investigate the Dirichlet boundary control of the Laplace equation, considering the control in $H^{1/2}(\partial \Omega)$, which is the natural space for Dirichlet data when the state belongs to $H^1(\Omega)$. The cost of the control is measured in the $H^{1/2}(\partial \Omega)$ norm that also plays the role of the regularization term. We discuss regularization and finite element error estimates enabling us to derive an optimal relation between the finite element mesh size $h$ and the regularization parameter $\varrho$, balancing the energy cost for the control and the accuracy of the approximation of the desired state. This relationship is also crucial in designing efficient solvers. We also discuss additional box constraints imposed on the control and the state. Our theoretical findings are complemented by numerical examples, including one example with box constraints.

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arXiv2025

State-based nested iteration solution of optimal control problems with PDE constraints

We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson equation discussed in this paper in detail but also parabolic and hyperbolic equations. The approach covers the standard $L^2$ setting as well as the more recent energy regularization, also including state and control constraints. We discretize OCPs subject to parabolic or hyperbolic PDEs by means of space-time finite elements similar as in the elliptic case. We discuss regularization and finite element error estimates, and derive an optimal relation between the regularization parameter and the finite element mesh size in order to balance the accuracy, and the energy costs for the corresponding control. Finally, we also discuss the efficient solution of the resulting systems of algebraic equations, and their use in a state-based nested iteration procedure that allows us to compute finite element approximations to the state and the control in asymptotically optimal complexity. The numerical results illustrate the theoretical findings quantitatively.

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arXiv2025

Inclusion of an Inverse Magnetic Hysteresis Model into the Space-Time Finite Element Method for Magnetoquasistatics

In this note we discuss the numerical solution of the eddy current approximation of the Maxwell equations using the simple Pragmatic Algebraic Model to include hysteresis effects. In addition to the more standard time-stepping approach we propose a space-time finite element method which allows both for parallelization and adaptivity simultaneously in space and time. Numerical experiments confirm both approaches yield the same numerical results.

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arXiv2025

An Air-Gap Element for the Isogeometric Space-Time-Simulation of Electric Machines

Space-time methods promise more efficient time-domain simulations, in particular of electrical machines. However, most approaches require the motion to be known in advance so that it can be included in the space-time mesh. To overcome this problem, this paper proposes to use the well-known air-gap element for the rotor-stator coupling of an isogeometric machine model. First, we derive the solution in the air-gap region and then employ it to couple the rotor and stator. This coupling is angle dependent and we show how to efficiently update the coupling matrices to a different angle, avoiding expensive quadrature. Finally, the resulting time-dependent problem is solved in a space-time setting. The spatial discretization using isogeometric analysis is particularly suitable for coupling via the air-gap element, as NURBS can exactly represent the geometry of the air-gap. Furthermore, the model including the air-gap element can be seamlessly transferred to the space-time setting. However, the air-gap element is well known in the literature. The originality of this work is the application to isogeometric analysis and space-time.

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arXiv2024

Efficient Solution of State-Constrained Distributed Parabolic Optimal Control Problems

We consider a space-time finite element method for the numerical solution of a distributed tracking-type optimal control problem subject to the heat equation with state constraints. The cost or regularization term is formulated in an anisotropic Sobolev norm for the state, and the optimal state is then characterized as the unique solution of a first kind variational inequality. We discuss an efficient realization of the anisotropic Sobolev norm in the case of a space-time tensor-product finite element mesh, and the iterative solution of the resulting discrete variational inequality by means of a semi-smooth Newton method, i.e., using an active set strategy.

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arXiv2024

Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems

We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state equation of the partial differential equation defines an isomorphism onto $H^{1,1/2}_{0;0,}(Q)$. Thus, we can eliminate the control from the tracking type functional to be minimized, to derive the optimality system in order to determine the state. Since the appearing operator induces an equivalent norm in $H_{0;0,}^{1,1/2}(Q)$, we will replace it by a computable realization of the anisotropic Sobolev norm, using a modified Hilbert transformation. We are then able to link the cost or regularization parameter $\varrho>0$ to the distance of the state and the desired target, solely depending on the regularity of the target. For a conforming space-time finite element discretization, this behavior carries over to the discrete setting, leading to an optimal choice $\varrho = h_x^2$ of the regularization parameter $\varrho$ to the spatial finite element mesh size $h_x$. Using a space-time tensor product mesh, error estimates for the distance of the computable state to the desired target are derived. The main advantage of this new approach is, that applying sparse factorization techniques, a solver of optimal, i.e., almost linear, complexity is proposed and analyzed. The theoretical results are complemented by numerical examples, including discontinuous and less regular targets. Moreover, this approach can be applied also to optimal control problems subject to non-linear state equations.

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arXiv2024

Robust finite element solvers for distributed hyperbolic optimal control problems

We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic distributed, tracking-type optimal control problems with both the standard $L^2$ and the more general energy regularizations. In contrast to the usual time-stepping approach, we discretize the optimality system by space-time continuous piecewise-linear finite element basis functions which are defined on fully unstructured simplicial meshes. If we aim at the asymptotically best approximation of the given desired state $y_d$ by the computed finite element state $y_{\varrho h}$, then the optimal choice of the regularization parameter $\varrho$ is linked to the space-time finite element mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L^2$ and the energy regularization, respectively. For this setting, we can construct robust (parallel) iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in the case of adaptive mesh refinements. The numerical results illustrate the theoretical findings firmly.

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arXiv2024

On a modified Hilbert transformation, the discrete inf-sup condition, and error estimates

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh parameter, but in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.

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arXiv2023

Parallel iterative solvers for discretized reduced optimality systems

We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the solution of elliptic tracking-type distributed optimal control problems with both the standard $L_2$ and the more general energy regularizations. If we aim at an approximation of the given desired state $y_d$ by the computed finite element state $y_h$ that asymptotically differs from $y_d$ in the order of the best $L_2$ approximation under acceptable costs for the control, then the optimal choice of the regularization parameter $\varrho$ is linked to the mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L_2$ and the energy regularization, respectively. For this setting, we can construct efficient parallel iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in case of adaptive mesh refinement. Similar results can be obtained for the space-time finite element discretization of the corresponding parabolic and hyperbolic optimal control problems.

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arXiv2023

Stable least-squares space-time boundary element methods for the wave equation

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $\operatorname{V}$ for the wave equation as a minimization problem in $L^2(Σ)$, where $Σ:= \partial Ω\times (0,T)$ is the lateral boundary of the space-time domain $Q := Ω\times (0,T)$. For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and, moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.

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arXiv2023

Adaptive least-squares space-time finite element methods

We consider the numerical solution of an abstract operator equation $Bu=f$ by using a least-squares approach. We assume that $B: X \to Y^*$ is an isomorphism, and that $A : Y \to Y^*$ implies a norm in $Y$, where $X$ and $Y$ are Hilbert spaces. The minimizer of the least-squares functional $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$, i.e., the solution of the operator equation, is then characterized by the gradient equation $Su=B^* A^{-1}f$ with an elliptic and self-adjoint operator $S:=B^* A^{-1} B : X \to X^*$. When introducing the adjoint $p = A^{-1}(f-Bu)$ we end up with a saddle point formulation to be solved numerically by using a mixed finite element method. Based on a discrete inf-sup stability condition we derive related a priori error estimates. While the adjoint $p$ is zero by construction, its approximation $p_h$ serves as a posteriori error indicator to drive an adaptive scheme when discretized appropriately. While this approach can be applied to rather general equations, here we consider second order linear partial differential equations, including the Poisson equation, the heat equation, and the wave equation, in order to demonstrate its potential, which allows to use almost arbitrary space-time finite element methods for the adaptive solution of time-dependent partial differential equations.

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arXiv2023

An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(Ω)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(Ω)$ involving a variable, mesh-dependent regularization parameter $\varrho(x)$. It turns out that the error between the computed finite element state $\widetilde{u}_{\varrho h}$ and the desired state $\overline{u}$ (target) is optimal in the $L^2(Ω)$ norm provided that $\varrho(x)$ behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm $\| \widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(Ω)}$ between the finite element state $\widetilde{u}_{\varrho h}$ and the target $\overline{u}$. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.

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arXiv2023

A space-time finite element method for the eddy current approximation of rotating electric machines

In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in space-time domain. Based on the Babu\v{s}ka--Ne\v{c}as theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability, efficiency and accuracy of the proposed approach.

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arXiv2023

Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints

In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common $L^2$ regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from wich we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.

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