Space-time Variational Formulation Research Articles

Computation of compressible and incompressible flows with a space–time stabilized finite element method

This paper presents the numerical results of compressible and incompressible flow problems through a unified approach based on a stabilized space–time finite element method. The numerical approach is continuous in space and discontinuous in time. The proposed method starts by the use of a stabilized space–time variational formulation, which allows the use of the same order interpolation functions for all solution variables. The numerical technique is tested through comparison with standard compressible and incompressible flow benchmarks. Compressible flow cases include 1D and 2D shock problems. Incompressible flow cases include lid-driven cavity flow and flow over a backward-facing step computed over a range of Re numbers. The results demonstrate high stability and accuracy of the numerical technique over a wide range of flow regimes, suggesting straightforward extension to many flow cases not yet investigated.

Fractional Space-Time Variational Formulations of (Navier--) Stokes Equations

Well-posed space-time variational formulations in fractional order Bochner--Sobolev spaces are proposed for parabolic partial differential equations, and in particular for the instationary Stokes and Navier--Stokes equations on bounded Lipschitz domains. The latter formulations include the pressure variable as a primal unknown and so account for the incompressibility constraint via a Lagrange multiplier. The proposed new variational formulations can be the basis of adaptive numerical solution methods that converge with the best possible rate, which, by exploiting the tensor product structure of a Bochner space, equals the rate of best approximation for the corresponding stationary problem. Unbounded time intervals are admissible in many cases, permitting an optimal adaptive solution of long-term evolution problems.

Aliasing Reduction with a 6th Order Space-Time Variational Formulation

Aliasing Reduction with a 6th Order Space-Time Variational Formulation

A reduced basis method for parabolic partial differential equations with parameter functions and application to option pricing

We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) as well as choosing the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time amd initial condition is introduced and shown to be stable. Finally, a Reduced Basis Method (RBM) is introduced that is able to use parameter functions also for the initial condition. Corresponding numerical results are shown.

Wavelet-In-Time Multigrid-In-Space Preconditioning of Parabolic Evolution Equations

Two space-time variational formulations of linear parabolic evolution equations are considered: one is symmetric and elliptic on the trial space, while the other is not. In each case, a space-time Petrov--Galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations. The well-posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis. For the iterative resolution of this algebraic system with parallelization in the temporal direction, we propose a sparse algebraic wavelet-in-time transformation on possibly nonuniform temporal meshes. This transformation approximately block-diagonalizes the preconditioner, and the individual spatial blocks can then be inverted, for instance, by standard spatial multigrid methods in parallel. The performance of the preconditioner is documented in a series of numerical experiments.

Acoustics in anisotropic poroelasticity: space–time variational formulation

A variational space–time formulation to treat acoustics problems in porous medium is developed. At first, no separation of the space and time has been applied in the approximation of the field variables. This produces a matrix variational formulation for more general numerical treatments. Finally, separation of variables has been applied to produce a simpler space–time formulation, in which the numerical treatment of the space domain and the time integration algorithms can be identified. The numerical treatment has been oriented to find symmetry in the space and time domains. Numerical simulation in the frequency domain and in the time domain have been performed to validate the variational formulation with separation of variables.

On Noncoercive Variational Inequalities

We consider variational inequalities with different trial and test spaces and a possibly noncoercive bilinear form. Well-posedness is shown under general conditions that are, e.g., valid for the space-time variational formulation of parabolic variational inequalities. Moreover, we prove an estimate for the error of a Petrov--Galerkin approximation in terms of the residual. For parabolic variational inequalities the arising estimate is independent of the final time.

An improved error bound for reduced basis approximation of linear parabolic problems

We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β δ \beta _{\delta } , the inverse of which enters into error estimates: β δ \beta _{\delta } is unity for the heat equation; β δ \beta _{\delta } decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.

A space–time variational approach to hydrodynamic stability theory

We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms ; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.

Treatment of fast moving loads in elastodynamic problems with a space-time variational formulation

The treatment of fast moving loads is faced with a variational space-time approach. A time integration algorithm and the variational treatment of fast moving loads are developed. The time integration algorithm and the treatment of moving loads are based on Hermite’s polynomials with independent field of velocities and discontinuous time derivative of the displacement field. The algorithm exhibits 6th order convergence and computational efficiency to treat problems of fast moving loads.

Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions

Space-time variational formulations and adaptive Wiener–Hermite polynomial chaos Galerkin discretizations of Kolmogorov equations in infinite dimensions, such as Fokker–Planck and Ornstein–Uhlenbeck equations for functions defined on an infinite-dimensional separable Hilbert space $$H$$ , are developed. The well-posedness of these equations in the Hilbert space $$\mathrm{L}^2(H,\mu )$$ of functions on the infinite-dimensional domain $$H$$ , which are square-integrable with respect to a Gaussian measure $$\mu $$ with trace class covariance operator $$Q$$ on $$H$$ , is proved. Specifically, for the infinite-dimensional Fokker–Planck equation, adaptive space-time Galerkin discretizations, based on a wavelet polynomial chaos Riesz basis obtained by tensorization of biorthogonal piecewise polynomial wavelet bases in time with a spatial Wiener–Hermite polynomial chaos arising from the Wiener–Itô decomposition of $$\mathrm{L}^2(H,\mu )$$ , are introduced. The resulting space-time adaptive Wiener–Hermite polynomial Galerkin discretization algorithms of the infinite-dimensional PDE are proved to converge quasioptimally in the sense that they produce sequences of finite-dimensional approximations that attain the best possible convergence rates afforded by best $$N$$ -term approximations of the solution from tensor-products of multiresolution (wavelet) time-discretizations and the Wiener–Hermite polynomial chaos in $$\mathrm{L}^2(H,\mu )$$ . As a consequence, the proposed adaptive Galerkin solution algorithms exhibit dimension-independent performance, which is optimal with respect to the algebraic best $$N$$ -term rate afforded by the solution and the polynomial degree and regularity of the multiresolution (wavelet) time-discretizations in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of “active” coordinates identified by the proposed adaptive Galerkin approximation algorithms. The computational work and memory required by the proposed algorithms scale linearly with the support size of the coefficient vectors that arise in the approximations, with dimension-independent constants.

A new error bound for reduced basis approximation of parabolic partial differential equations

We consider a space–time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov–Galerkin truth finite element discretization with favorable discrete inf-sup constant βδ: βδ is unity for the heat equation; βδ grows only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates.

Space-Time Reduced Basis Methods for Time-Periodic Partial Differential Equations

AbstractWe consider parameter-dependent time-periodic parabolic problems as they occur e.g. in the modeling of rotating propellers. The goal is to determine the parameters (e.g. design or steering parameters) such that a given output of interest (e.g. the efficiency of the propeller) is maximized. A standard approach to numerically solve such problems is a time-stepping (fixed-point) scheme. This approach, however, often suffers from some drawbacks, in particular within the Reduced Basis Method (RBM). First, there might be long transient phases before a periodic or steady state is reached, which is particularly disadvantageous in the online phase. Moreover, corresponding error estimates usually include sums over time-steps which might become inaccurate.Instead, we consider a space-time variational formulation using periodic basis functions in time, which avoids the need for fixed-point iterations. Based on this variational formulation, we develop a space-time RBM using wavelets in time and derive corresponding a-posteriori error estimates. We present numerical results indicating the efficiency of the method as well as the effectivity of the derived error bounds.

Divergence-free wavelet bases on the hypercube: Free-slip boundary conditions, and applications for solving the instationary Stokes equations

We construct wavelet Riesz bases for the usual Sobolev spaces of divergence free functions on that have vanishing normals at the boundary. We give a simultaneous space-time variational formulation of the instationary Stokes equations that defines a boundedly invertible mapping between a Bochner space and the dual of another Bochner space. By equipping these Bochner spaces by tensor products of temporal and divergence-free spatial wavelets, the Stokes problem is rewritten as an equivalent well-posed bi-infinite matrix vector equation. This equation can be solved with an adaptive wavelet method in linear complexity with best possible rate, that, under some mild Besov smoothness conditions, is nearly independent of the space dimension. For proving one of the intermediate results, we construct an eigenfunction basis of the stationary Stokes operator.

Adaptive Wavelet Schemes for Parabolic Problems: Sparse Matrices and Numerical Results

A simultaneous space-time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method. This method is shown to converge with the best possible rate in linear complexity. Thanks to the use of tensor product bases, there is no penalty in complexity due to the additional time dimension. Special wavelets are designed such that the bi-infinite system matrix is sparse. This sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method. Numerical results for an ODE and the heat equation are presented.

An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations

In this paper we propose a discontinuous Galerkin scheme for the numerical approximation of unsteady heat conduction and diffusion problems in multi dimensions. The scheme is based on a discrete space–time variational formulation and uses an explicit approximative solution as predictor. This predictor is obtained by a Taylor expansion about the barycenter of each grid cell at the old time level in which all time or mixed space–time derivatives are replaced by space derivatives using the differential equation several times. The heat flux between adjacent grid cells is approximated by a local analytical solution. It takes into account that the approximate solution may be discontinuous at grid cell interfaces and allows the approximation of discontinuities in the heat conduction coefficient. The presented explicit scheme has to satisfy a typical parabolic stability restriction. The loss of efficiency, especially in the case of strongly varying sizes of cells in unstructured grids, is circumvented by allowing different time steps in each grid cell which are adopted to the local stability restrictions. We discuss the linear stability properties in this case of varying diffusion coefficients, varying space increments and local time steps and extent these considerations also to a modified symmetric interior penalization scheme. In numerical simulations we show the efficiency and the optimal order of convergence in space and time.

A space-time variational formulation for the boundary integral equation in a 2D elastic crack problem

— This paper investigates the transient elastic wave scattering by a crack in R by means of Boundary Integral Equation Method. The analysis of the Laplace-Fourier transform (in time) of the integral operator allows to obtain existence, uniqueness and continuity dependence of the solution with respect to the data, in a Sobolev functional framework. A regularisation of the hypersingular BIE is applied in order to remove the hypersingularity and to write the associated time-space variational formulation on a tractable form. A Galerkin-type approximation is then performed to solve this variational formulation and we finally present some numerical results. Resume. — Nous nous interessons a la resolution par une methode d'equations integrales d'un probleme de diffraction d'ondes elastiques transitoires par une fissure. L'analyse de la transformee de F ourier-Laplace (en temps) de Voperateur integral permet d'obtenir des resultats d'existence, d'unicite et de continuite de la solution par rapport aux donnees dans des espaces fonctionnels de type Sobolev. Le noyau de V equation integrale etant hypersingulier, on ne dispose pas directement d'une expression calculable de la forme bilineaire associee. Cette difficulte est surmontee en appliquant une methode de regularisation qui fournit finalement la formulation variationnelle espace-temps du probleme. Cette formulation est enfin approchee a l'aide d'un schema de Galerkin et nous presentons quelques resultats numeriques.