Space-time Finite Element Discretization Research Articles

Inverse problems for linear hyperbolic equations using mixed formulations

We introduce a direct method which allows the solving of numerically inverse problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the equation posed in —Ω being a bounded subset of —from a partial distributed observation. We employ a least-squares technique and minimize the L2-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf–sup condition) and then introduce a numerical approximation based on space-time finite element discretization. We prove the strong convergence of the approximation and then discuss several examples for N = 1 and N = 2. The problem of the reconstruction of both the state and the source terms is also addressed.

A space-time hp-interpolation-based certified reduced basis method for Burgers' equation

We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases.

An Eulerian Space-Time Finite Element Method for Diffusion Problems on Evolving Surfaces

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in $\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-time continuum $\Bbb{R}^{d+1}$. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stability results are derived. The analysis is based on an inf-sup condition and involves some natural, but nonstandard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization. This DG space-time method is explained and results of numerical experiments are presented that illustrate its properties.

Indirect Multiple Shooting for Nonlinear Parabolic Optimal Control Problems with Control Constraints

We discuss the indirect multiple shooting approach for the solution of PDE-based parabolic optimal control problems with control constraints. The method is formulated within an abstract function space setting and uses a space-time Galerkin finite element discretization. The emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the PDE context. Numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail.

Stability of sparse space-time finite element discretizations of linear parabolic evolution equations

The abstract linear parabolic evolution equation is formulated as a well-posed linear operator equation for which a conforming minimal residual Petrov–Galerkin discretization framework is developed: the approximate solution is defined as the minimizer of a suitable functional residual over the discrete test space, and may be obtained numerically from an equivalent algebraic residual minimization problem. This approximate solution is shown to be well defined and to converge quasi-optimally in the natural norm if the discrete trial and test spaces are stable, i.e., if the discrete inf–sup condition is satisfied with a uniform positive lower bound. For the parabolic operator we devise an abstract criterion for the stability of pairs of space–time trial and test spaces, and construct hierarchic families of trial and test spaces of a sparse space–time tensor-product type that satisfy this criterion. The theory is applied to the concrete example of the diffusion equation and is illustrated numerically.

Goal‐oriented space–time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow

SUMMARYThis paper presents a general strategy for designing adaptive space–time finite element discretizations of the nonstationary Navier–Stokes equations. The underlying framework is that of the dual weighted residual method for goal‐oriented a posteriori error estimation and automatic mesh adaptation. In this approach, the error in the approximation of certain quantities of physical interest, such as the drag coefficient, is estimated in terms of local residuals of the computed solution multiplied by sensitivity factors, which are obtained by numerically solving an associated dual problem. In the resulting local error indicators, the effects of spatial and temporal discretization are separated, which allows for the simultaneous adjustment of time step and spatial mesh size. The efficiency of the proposed method for the construction of economical meshes and the quantitative assessment of the error is illustrated by several test examples. Copyright © 2012 John Wiley & Sons, Ltd.

Space-time finite element approximation of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. Furthermore, we also propose a space-time mixed finite element discretization scheme to approximate the space-time elliptic equations, and derive a priori error estimates for the state and the adjoint state. Numerical examples are presented to illustrate our theoretical findings and the performance of our approach.

Adaptive Finite Element Methods for Optimal Control of Elastic Waves

AbstractIn this paper a posteriori error estimates for space-time finite element discretizations for optimal control problems governed by the dynamical Lamé system are considered using the dual weighted residual method (DWR). We apply techniques developed in Kröner (2011a), where optimal control problems for second order hyperbolic equations are considered. The provided error estimator separates the influences of different parts of the discretization (time, space, and control discretization). This allows us to set up an adaptive algorithm which improves the accuracy of the computed solutions by construction of locally refined meshes. We present a numerical example showing a speedup in cpu-time as well as a reduction in degrees of freedom in comparison to uniform mesh refinement.

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

In this paper, a priori error estimates for space–time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from Meidner and Vexler (SIAM Control Optim 47(3):1150–1177, 2008; SIAM Control Optim 47(3):1301–1329, 2008) where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches, e.g., cellwise linear discretization in space, the postprocessing approach from Meyer and Rosch (SIAM J Control Optim 43:970–985, 2004), and the variationally discrete approach from Hinze (J Comput Optim Appl 30:45–63, 2005) can be obtained. In addition, we derive an estimate for a setting with finitely many time-dependent controls.

Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

In this paper optimal control problems governed by the wave equation with control constraints are analyzed. Three types of control action are considered: distributed control, Neumann boundary control, and Dirichlet control, and proper functional analytic settings for them are discussed. For treating inequality constraints, semismooth Newton methods are discussed and their convergence properties are investigated. In the case of distributed and Neumann control, superlinear convergence is shown. For Dirichlet boundary control, superlinear convergence is proved for a strongly damped wave equation. For numerical realization, a space-time finite element discretization is discussed. Numerical examples illustrate the results.

A Priori Error Estimates for Finite Element Discretizations of Parabolic Optimization Problems with Pointwise State Constraints in Time

In this paper, we consider an optimal control problem which is governed by a linear parabolic equation and is subject to state constraints pointwise in time. Optimal order error estimates are developed for a space-time finite element discretization of this problem. Numerical examples confirm the theoretical results. As a by-product of our analysis, we derive a new regularity result for the optimal control.

Adaptive Finite Element Methods For Optimal Control Of Second Order Hyperbolic Equations

Abstract In this paper we consider a posteriori error estimates for space-time finite element discretizations for optimal control of hyperbolic partial dierential equations of second order. It is an extension of Meidner and Vexler (2007), where optimal control problems of parabolic equations are analyzed. The state equation is formulated as a first order system in time and a posteriori error estimates are derived separating the in uences of time, space, and control discretization. Using this information the accuracy of the solution is improved by local mesh refinement. Numerical examples are presented. Finally, we analyze the conservation of energy of the homogeneous wave equation with respect to dynamically in time changing spatial meshes.

A posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations

We present a novel a posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations. Our analysis is based on the equivalence of error and residual and a suitable decomposition of the residual into spatial and temporal contributions. In contrast to existing results we directly bound the error of the full space-time discretization and do not resort to auxiliary semi-discretizations. We thus obtain sharper bounds. Moreover the present analysis covers a wider range of discretizations both with respect to time and to space.

A multiscale finite element space–time discretization method for transient transport phenomena using bubble functions

A space–time finite element discretization method for unsteady transport phenomena is formulated using a variational multiscale finite element scheme. The described discretization is based on the utilization of bubble function enriched finite elements. The scheme is applied to model unsteady diffusion and convection–diffusion equations. It is shown that any temporal and spatial instabilities can be removed by appropriate enrichment of linear Lagrangian finite element approximations in the context of the standard Galerkin method. The proposed scheme is compared with widely used θ time stepping method. Numerical results generated by the proposed scheme are validated via their comparison with the analytical solution of a bench mark problem.

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments.

Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations

In this paper, we develop an error estimator and an adaptive algorithm for efficient solution of parabolic partial differential equations. The error estimator assesses the discretization error with respect to a given quantity of physical interest and separates the influence of the time and space discretizations. This allows us to set up an efficient adaptive strategy producing economical (locally) refined meshes for each time step and an adapted time discretization. The space and time discretization errors are equilibrated, leading to an efficient method.

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part II: Problems with Control Constraints

This paper is the second part of our work on a priori error analysis for finite element discretizations of parabolic optimal control problems. In the first part [SIAM J. Control Optim., 47 (2008), pp. 1150–1177] problems without control constraints were considered. In this paper we derive a priori error estimates for space-time finite element discretizations of parabolic optimal control problems with pointwise inequality constraints on the control variable. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For the treatment of the control discretization we discuss different approaches, extending techniques known from the elliptic case.

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

AbstractThe paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects.The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.

Efficient numerical solution of parabolic optimization problems by finite element methods

We present an approach for efficient numerical solution of optimization problems governed by parabolic partial differential equations. The main ingredients are: space-time finite element discretization, second-order optimization algorithms, and storage reduction techniques. We discuss the combination of these components for the solution of large-scale optimization problems.

Optimal Vortex Reduction for Instationary Flows Based on Translation Invariant Cost Functionals

We consider the problem of an appropriate choice of a cost functional for vortex reduction for unsteady flows described by the Navier–Stokes equations. This choice is directly related to a physically correct definition of a vortex. Therefore, we discuss different possibilities for the cost functional and analyze the resulting optimal control problems. Moreover, we present an efficient numerical realization of this concept based on space-time finite element discretization and demonstrate its behavior in some numerical experiments. It is demonstrated that the choice of cost functionals has a significant effect on the reduction of vortices.