Time Discontinuous Galerkin Method Research Articles

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

Abstract We first give a general error estimate for the nonconforming approximation of a problem for which a Banach–Nečas–Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $\theta -$scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $\theta \le \frac 1 2$. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

A comparison study of spatial and temporal schemes for flow and transport problems in fractured media with large parameter contrasts on small length scales

AbstractIn this work, various high-accuracy numerical schemes for transport problems in fractured media are further developed and compared. Specifically, to capture sharp gradients and abrupt changes in time, schemes with low order of accuracy are not always sufficient. To this end, discontinuous Galerkin up to order two, Streamline Upwind Petrov-Galerkin, and finite differences, are formulated. The resulting schemes are solved with sparse direct numerical solvers. Moreover, time discontinuous Galerkin methods of order one and two are solved monolithically and in a decoupled fashion, respectively, employing finite elements in space on locally refined meshes. Our algorithmic developments are substantiated with one regular fracture network and several further configurations in fractured media with large parameter contrasts on small length scales. Therein, the evaluation of the numerical schemes and implementations focuses on three key aspects, namely accuracy, monotonicity, and computational costs.

Error estimates for finite element discretizations of the instationary Navier–Stokes equations

In this work we consider the two dimensional instationary Navier–Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the L∞(I; L2(Ω)), L2(I; H1(Ω)) and L2(I; L2(Ω)) norms have been shown. The main result of the present work extends the error estimate in the L∞(I; L2(Ω)) norm to the Navier–Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the L∞(I; L2(Ω)) error estimate, also allow us to show best approximation type error estimates in the L2(I; H1(Ω)) and L2(I; L2(Ω)) norms, which complement this work.

Post-processing and improved error estimates of numerical methods for evolutionary systems

Abstract We consider evolutionary systems, i.e., systems of linear partial differential equations arising from the mathematical physics. For these systems, there exists a general solution theory in exponentially weighted spaces, which can be exploited in the analysis of numerical methods. The numerical method considered in this paper is a discontinuous Galerkin method in time combined with a conforming Galerkin method in space. Building on our recent paper (Franz, S., Trostorff, S. & Waurick, M. (2019) Numerical methods for changing type systems. IMAJNA, 39, 1009–1038), we improve some of the results, study the dependence of the numerical solution on the weight parameter and consider a reformulation and post-processing of its numerical solution. As a by-product, we provide error estimates for the dG-C0 method. Numerical simulations support the theoretical findings.

Sparse grid time-discontinuous Galerkin method with streamline diffusion for transport equations

High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. Following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. By the growth estimates, error estimates are derived with non-smooth initial data. In particular, graded temporal grids are used to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results.

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

A Novel Space-Time Finite Element Algorithm to Investigate the Hygro-Mechanical Behaviours of Wood Fiber-Polymer Composites

Moisture-induced swelling, or hygroexpansion, has been known to greatly deteriorate the durability of wood fiber-polymer composites (WPC). It is generally very expensive to perform experiments to completely obtain the diffusion kinetics as the process occurs in a very extended period of time. For the first time, we have developed a space-time finite element algorithm that employs time discontinuous Galerkin (TDG) method for time-dependent 3D hygro-mechanical behaviours of WPC. The formulation of matrix equations in spatial and temporal domains are explained in detail. A block Gauss-Seidel iterative method is used in the predictor/multi-corrector multi-pass algorithm, which efficiently yields unconditionally stable and high-order accurate solutions. The model is validated by comparing the predicted time-dependent hygroexpansion with that obtained in a previous experimental study. The quantitative analysis ensures the reliability of model, based on a Fickian diffusion process. With our adaptive time-stepping scheme that bases on the embedded solution from the multi-pass iterations, the model efficiently progresses the kinetics with relatively large time steps. A runtime of a few hours compared to about three months of actual laboratory experimentation confirms the novelty and robustness of our model.

Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation

<abstract><p>In this paper, a fully discrete local discontinuous Galerkin finite element method is proposed to solve the KdV-Burgers-Kuramoto equation with variable-order Riemann-Liouville time fractional derivative. The method proposed in this paper is based on the finite difference method in time and local discontinuous Galerkin method in space. For all $ \epsilon(t)\in (0, 1) $ with variable order, we prove the scheme is unconditional stable and convergent. Finally, numerical examples are provided to verify the theoretical analysis and the order of convergence for the proposed method.</p></abstract>

A priorierror estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

In this paper we investigatea priorierror estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.

Temporally Semidiscrete Approximation of a Dirichlet Boundary Control for a Fractional/Normal Evolution Equation with a Final Observation

Optimal Dirichlet boundary control for a fractional/normal evolution equation with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The convergence of a temporally semidiscrete approximation is rigorously established, where the control is not explicitly discretized and the state equation is discretized by a discontinuous Galerkin method in time. Numerical results are provided to verify the theoretical results.

A space-time finite element method based on local projection stabilization in space and discontinuous Galerkin method in time for convection-diffusion-reaction equations

In this article, we combine the local projection stabilization (LPS) technique in space and the discontinuous Galerkin (DG) method in time to investigate the time-dependent convection-diffusion-reaction problems. This kind of stabilized space-time finite element (STFE) scheme, based on approximation space enriched by bubble functions that can increase stability, is constructed. The existence, uniqueness and stability are proved with the properties of Lagrange interpolation polynomials established on Radau points in time direction. An error estimate in L∞(L2)-norm is given by introducing the elliptic projection operators in space direction. This estimate approach is different from the previous ones that construct a special interpolant into approximation space showing an extra orthogonality property on the projection space. Since the techniques of Lagrange interpolation in time direction decouple time and space variables, the method proposed in this paper has the advantages of reducing calculation and simplifying theoretical analysis. The space and time convergence orders are illustrated in the first numerical example with smooth solutions. A comparison between the traditional STFE scheme and the constructed scheme for the problem having exponential boundary layers is presented in the second numerical example. The simulation results show that the novel method can greatly reduce nonphysical oscillations. And the influences of the stabilization parameters on the behavior of the approximate solution are discussed by some numerical results.

A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system

<p style='text-indent:20px;'>In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.</p>

Space–time domain decomposition for two-phase flow between different rock types

In Ahmed et al. (2019) a space–time domain decomposition method was proposed for two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error.In this paper we provide a mathematical and numerical analysis of this space–time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase flow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on the solution of space–time nonlinear subdomain problems over the whole time interval, allowing for different time steps in different parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method.

Discontinuous Galerkin Method and Dual-SLERP for Time Integration of Flexible Multibody Dynamics

Abstract A novel time-discontinuous Galerkin (DG) method is introduced for the time integration of the differential-algebraic equations governing the dynamic response of flexible multibody systems. In contrast to traditional Galerkin methods, the rigid-body motion field is interpolated using the dual spherical linear scheme. Furthermore, the jumps inherent to time-DG methods are expressed in terms of a parameterization of the relative motion from one time-step to the next. The proposed scheme is third-order accurate for initial value problems of both rigid and flexible multibody dynamics.

Numerical approximation of time-fractional Burgers-type equation

In this work, we analyze and test a local discontinuous Galerkin method for solving the Burgers-type equation. The proposed numerical method, which is high-order accurate, is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We prove that the scheme is unconditionally stable and convergent. Some numerical tests are provided to illustrate the accuracy and capability of the scheme.

Local projection stabilization with discontinuous Galerkin method in time applied to convection dominated problems in time-dependent domains

This paper presents the numerical analysis of a stabilized finite element scheme with discontinuous Galerkin (dG) discretization in time for the solution of a transient convection–diffusion–reaction equation in time-dependent domains. In particular, the local projection stabilization and the higher order dG time stepping scheme are used for convection dominated problems. Further, an arbitrary Lagrangian–Eulerian formulation is used to handle the time-dependent domain. The stability and error estimates are given for the proposed numerical scheme. The validation of the proposed local projection stabilization scheme with higher order dG time discretization is demonstrated with appropriate numerical examples.

A time discontinuous Galerkin isogeometric analysis method for non-Fourier thermal wave propagation problem

PurposeNumerical instability such as spurious oscillation is an important problem in the simulation of heat wave propagation. The purpose of this study is to propose a time discontinuous Galerkin isogeometric analysis method to reduce numerical instability of heat wave propagation in the medium subjected to heat sources, particularly heat impulse.Design/methodology/approachThe essential vectors of temperature and the temporal gradients are assumed to be discontinuous and interpolated individually in the discretized time domain. The isogeometric analysis method is applied to use its property of smooth description of the geometry and to eliminate the mesh-dependency. An artificial damping scheme with proportional stiffness matrix is brought into the final discretized form to reduce the numerical spurious oscillations.FindingsThe numerical spurious oscillations in the simulation of heat wave propagation are effectively eliminated. The smooth description of geometry with spline functions solves the mesh-dependency problem and improves the numerical precision.Originality/valueThe time discontinuous Galerkin method is applied within the isogeometric analysis framework. The proposed method is effective in the simulation of the wave propagation problems subjecting to impulse load with numerical stability and accuracy.

Space–time Galerkin methods for simulation of laser heating using the generalized nonlinear model

The generalized thermal model is a thermodynamically consistent extension of the classical Fourier’s law for describing thermal energy transportation which is very relevant to applications involving very small length, time scales and/or at extremely low temperatures. Under such conditions, thermal propagation has been observed to manifest as waves, a phenomenon widely referred to as second sound effect. However, this is in contrast to the paradoxical prediction of the Fourier’s model that thermal disturbances propagate with infinite speed. In this work, we review the nonlinear model based on the theory of Green and Naghdi for thermal conduction in rigid bodies and present its implementation within a class of space–time methods. The unconditional stability of the time-discontinuous Galerkin method without restriction over the grid structure of the space–time domain is proved. We also perform a number of numerical experiments to study the convergence properties and analyze the thermal response of materials under short-pulsed laser heating in two space dimensions.

Optimal control problem of the two-dimensional modified anomalous subdiffusion equation with discontinuous Galerkin approximation

In this paper, we consider an optimal control problem (OCP) governed by a modified anomalous subdiffusion equation with box constraints on the control. We apply space–time discontinuous Galerkin method to discretize the problem. Specifically, symmetric interior penalty Galerkin method and piecewise discontinuous constant temporal approximations are used. Then, we discuss the equivalence of optimize then discretize and discretize then optimize approaches in a general setting. We derive stability and convergence analysis for the fully discrete problem. At the end, we present some numerical results to measure the numerical rate of convergence.