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.

In this paper we formulate and analyze adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative v . del u is considered as part of the total time derivative d/dt u = partial derivative(t)u + v . del 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-0(1)(Omega) such that the convective derivative is considered as an element of the dual H-1(Omega) of the test space H-0(1)(Omega), which also allows unbounded velocities 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.

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.

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 $L^2(\Sigma)$, where $\Sigma := \partial \Omega \times (0,T)$ is the lateral boundary of the space-time domain $Q := \Omega \times (0,T)$. 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.

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.

We investigate the Dirichlet boundary control of the Laplace equation, considering the control in H^1/2(∂ Ω), which is the natural space for Dirichlet data when the state belongs to H^1(Ω). The cost of the control is measured in the H^1/2(∂ Ω) 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 ϱ, 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.

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.

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.

In this note we formulate and analyze a hybrid space-time finite element method for the numerical solution of parabolic evolution equations. We combine the more standard variational formulation in Bochner spaces, and a more recent formulation in anisotropic Sobolev spaces using a modified Hilbert transformation. The Galerkin discretization then results in symmetric and positive definite stiffness matrices for both the temporal and spatial derivatives, and a remainder which is in general non-symmetric, but non-negative. We present related error estimates a series of different numerical examples which confirm the theoretical findings.

In this paper we formulate and analyze a least squares boundary element method for the weakly singular boundary integral equation which is related to the solution of a Dirichlet boundary value problem for a second order partial differential equation, with the Laplacian as model problem. In particular we may assume less regular boundary data g ∉H^1/2(Γ ) but g ∈ L^2(Γ ) . For this we consider the single layer boundary integral operator V: H^-1(Γ ) → L^2(Γ ) , i.e., we will solve the boundary integral equation Vw=f by minimizing 1/2 ‖ V w - f ‖ _L^2(Γ )^2 . This results in a mixed variational formulation where we use piecewise constant approximations to discretize both the primal unknown w ∈ H^-1(Γ ) and the adjoint p:=f-Vw ∈ L^2(Γ ) . Using nested boundary element spaces S_H^0(Γ ) ⊆ S_h^0(Γ ) we can prove stability and related error estimates for both the primal and adjoint approximations, w_H and p_h , respectively. When considering the approximate adjoint p_h on a finer mesh than the primal w_H , we can use ‖ p_h ‖ _L^2(Γ ) as a posteriori error indicator for the error ‖ w-w_H ‖ _H^-1(Γ ) to drive an adaptive mesh refinement. Note that this defines an adaptive boundary element method also for regular boundary data g ∈ H^1/2(Γ ) . Numerical examples confirm the theoretical results.

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.

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.

Purpose 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. The purpose of this paper is to overcome this problem by proposing the use of the well-known air-gap element for rotor-stator coupling of an isogeometric machine model. Design/methodology/approach First, the authors derive the solution in the air-gap region and then use it to couple the rotor and stator. This coupling is angle dependent and the authors 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. Findings Spatial discretization using isogeometric analysis is particularly suitable for coupling via the air-gap element, as nonuniform rational B-splines 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. Originality/value The air-gap element is well known in the literature. The originality of this work is the application to isogeometric analysis and space-time.

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 L2 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 yd by the computed finite element state yϱh, then the optimal choice of the regularization parameter ϱ is linked to the space-time finite element mesh-size h by the relations ϱ=h4 and ϱ=h2 for the L2 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.

We consider space-time tracking type distributed optimal control problems for the wave equation in the space-time domain $Q:= \Omega \times (0,T) \subset {\mathbb{R}}^{n+1}$, where the control is assumed to be in the energy space $[H_{0;,0}^{1,1}(Q)]^*$, rather than in $L^2(Q)$ which is more common. While the latter ensures a unique state in the Sobolev space $H^{1,1}_{0;0,}(Q)$, this does not define a solution isomorphism. Hence we use an appropriate state space $X$ such that the wave operator becomes an isomorphism from $X$ onto $[H_{0;,0}^{1,1}(Q)]^*$. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^2(Q)}$ between the computed space-time finite element solution $\widetilde{u}_{\varrho h}$ and the target function $\overline{u}$ with respect to the regularization parameter $\varrho$, and the space-time finite element mesh-size $h$, depending on the regularity of the desired state $\overline{u}$. These estimates lead to the optimal choice $\varrho=h^2$ in order to define the regularization parameter $\varrho$ for a given space-time finite element mesh size $h$, or to determine the required mesh size $h$ when $\varrho$ is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.

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 cS depends linearly on the finite element mesh size h. While the ratio cS/h decreases as 1/T for T→∞, numerical results indicate a decay of cS/h≃ν−α for some α∈[14,13] in the polynomial degree ν of the finite element basis functions. However, in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.

We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babu\v{s}ka's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.

Abstract Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p ∈ L 2 ⁢ ( Ω ) {p\in L^{2}(\Omega)} . The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual L2(Ω) norm regularization term with a constant regularization parameter ϱ is replaced by a suitable representation of the energy norm in H−1(Ω) involving a variable, mesh-dependent regularization parameter ϱ(x). It turns out that the error between the computed finite element state u˜ϱh and the desired state u‾ (target) is optimal in the L2(Ω) norm provided that ϱ(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 ‖u˜ϱh−u‾‖L2(Ω) between the finite element state u˜ϱh and the target 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.

Shape and topology optimization of electrical machines [3] as well as the optimal control [4] subject to parabolic evolution equations require an efficient solution of the direct simulation problem which is forward in time, and in most cases also of the adjoint problem which is backward in time.