Space-time Finite Element Research Articles

High-order Mass- and Energy-conserving SAV-Gauss Collocation Finite Element Methods for the Nonlinear Schrödinger Equation

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form $O(h^p+\tau^{k+1})$ in the $L^\infty(0,T;H^1)$-norm is established, where $h$ and $\tau$ denote the spatial and temporal mesh sizes, respectively, and $(p,k)$ is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.

Space-Time Finite Element Discretization of Parabolic Optimal Control Problems with Energy Regularization

In this paper, 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 Babuška'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.

Unstructured Space-Time Finite Element Methods for Optimal Control of Parabolic Equations

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, which is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residual-type error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.

Determination of the stress-strain state of structures made of viscoelastic fibrous composite materials based on the stiffness matrix of a spatio-temporal finite element

Abstract. Increase in spheres of composite materials application leads to the need to improve the methods of composite structures calculation, increase their efficiency and adequacy to real processes. The use of fibrous composite materials creates difficulties in determining the stress-strain state of the structure due to the different physical and mechanical properties of the material components. As a rule, a fibrous composite consists of two components: a matrix and a fiber. The matrix ensures the material monolithicity fixing the shape of the product and the relative position of the reinforcing fibers. The fibers absorb the main stresses that occur during operation, and provide rigidity and strength of the composite. The combination of the matrix and fiber properties allows creating special types of composites to be used in various equipment. The use of rubber as a matrix gives the structure viscoelastic properties, which can be seen in the deformation creeping and stress relaxation in the material, which complicates the calculation of the stress-strain state of the whole structure greatly. The paper proposes an approach that is the development of a moment scheme of finite elements in the form of the displacements method based on the Lagrange principle. Using this approach, a special space-time finite element has been developed that takes into account the viscoelastic properties of the material matrix and the fiber elastic properties. The proposed numerical approach allows obtaining a solution to the problems of viscoelasticity of composite materials mechanics in three-dimensional model. Using the applications package created on the basis of this approach, the calculation of the stress-strain state of a single-cavity pneumatic cylinder with a rubber cord shell has been done. The influence of the reinforcement scheme and rubber viscoelastic properties on the parameters of the pneumatic cylinder deformation has been analyzed. In particular, it becomes possible to adjust the structure stiffness properties by changing the orientation, type and concentration of fibers, as well as the type of matrix and other material characteristics. Keywords: fibrous composite material, viscoelasticity, space-time finite element.

Efficient implementation of space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows

The present work is devoted to the study of efficient implementation of spacetime adaptive ADER finite element discontinuous Galerkin method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter scheme for simulation of non-stationary compressible multicomponent reactive flows. The multicomponent and reaction properties of the flow are considered in the form of convection-reaction equations. Therefore an effective scheme of splitting the original nonlinear system of algebraic equations of LST-DG predictor was developed to obtain an efficient iterative solution method. This approach is based on the use of the linearity property of the system of equations for the discrete spacetime solution for the system of convection equations, which makes it possible to pre-compute the inverse matrices, and allows the vectorized calculation of the transfer of individual components. The change in the concentrations of the components in the convection-reaction equations and the energy yield in the gas-dynamic equations due to the reactions are calculated using the global iterations. The iterative scheme for the original essentially nonlinear system of equations is split into a sequence of smaller systems. The approach is well applicable for the problems, especially in the case of a large number of components and reactions in the flow.

Further results on a space-time FOSLS formulation of parabolic PDEs

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer and Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.

Space-Time-Parallel 3-D Finite Element Transformer Model With Adaptive TLM and Parareal Techniques for Electromagnetic Transient Analysis

In this paper, a novel massively space-time parallel finite element (FE) transformer model is proposed for power system electromagnetic transient simulations. The method utilizes the reduced magnetic vector potential (RMVP) formulation to solve the electromagnetic field in the transformer, and the 3-D tetrahedral edge element is applied to discretize the RMVP formulation. The discretized nonlinear system is solved by adaptive transmission line modeling (TLM) and matrix-free conjugate gradient iterator, which allows parallelism at the elemental-level and eliminates the need for Newton-Raphson (NR) iterations over the large-scale global matrix. The paper also integrates the parallel-in-time method with the finite element solver to further parallelize the model at space-time level, and the GPU implementation of the model realized a speedup of over 55 and high accuracy, compared with a commercial FE software.

An adjoint-based a posteriori analysis of numerical approximation of Richards equation

<p style='text-indent:20px;'>This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.</p>

Dynamic Analysis of Fill Dam Using Velocity-Based Space-Time Finite Element Method with Elasto-Plastic Constitutive Model

Dynamic Analysis of Fill Dam Using Velocity-Based Space-Time Finite Element Method with Elasto-Plastic Constitutive Model

An Exact Realization of a Modified Hilbert Transformation for Space-Time Methods for Parabolic Evolution Equations

Abstract We present different possibilities of realizing a modified Hilbert type transformation as it is used for Galerkin–Bubnov discretizations of space-time variational formulations for parabolic evolution equations in anisotropic Sobolev spaces of spatial order 1 and temporal order 1 2 \frac{1}{2} . First, we investigate the series expansion of the definition of the modified Hilbert transformation, where the truncation parameter has to be adapted to the mesh size. Second, we introduce a new series expansion based on the Legendre chi function to calculate the corresponding matrices for piecewise polynomial functions. With this new procedure, the matrix entries for a space-time finite element method for parabolic evolution equations are computable to machine precision independently of the mesh size. Numerical results conclude this work.

Finite element approximation of time fractional optimal control problem with integral state constraint

In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation

In this paper we investigate in detail the applications of the classical Newton-Raphson method in connection with a space-time finite element discretization scheme for the inviscid Burgers equation in one dimensional space. The underlying discretization method is the so-called streamline diffusion method, which combines good stability properties with high accuracy. The coupled nonlinear algebraic equations thus obtained in each space-time slab are solved by the generalized Newton-Raphson method. Exploiting the band-structured properties of the Jacobian matrix, two different algorithms based on the Newton-Raphson linearization are proposed. In a series of examples, we show that in each time-step a quadratic convergence order is attained when the Newton-Raphson procedure applied to the corresponding system of nonlinear equations.

Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources

Abstract We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.

Space-time finite element method for determination of a source in parabolic equations from boundary observations

Abstract A novel inverse source problem concerning the determination of a term in the right-hand side of parabolic equations from boundary observation is investigated. The observation is given by an imprecise Dirichlet data on some part of the boundary. The unknown heat source is sought as a function depending on both space and time variables with an a priori information. The problem is reformulated as an optimal control problem with a Tikhonov regularization term. The gradient of the functional is derived via an adjoint problem. The space-time discretization approach is employed which allows the use of general space-time finite elements. The convergence of the approach is proved. Some numerical examples are presented for showing the efficiency of the approach.

Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation

Abstract This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The state equation is discretized by a space-time finite element method. The controls are not discretized. Under suitable assumptions optimal convergence rates for the error in the state and control variable are proven. Based on a conditional gradient method the solution of the semi-discretized optimal control problem is computed. The theoretical convergence rates are confirmed in a numerical example.

A note on the efficient evaluation of a modified Hilbert transformation

Abstract In this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H 1,1/2(Q). The resulting bilinear form of the first order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.

Convergence of finite element methods for hyperbolic heat conduction model with an interface

The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures.

Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations

Abstract We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

Space–time FEM with block-iterative algorithm for nonlinear dynamic fracture analysis of concrete gravity dam

In this paper, space–time finite element method is developed for the dynamic fracture analysis of dam–reservoir (DR) system which is supported on a perfectly rigid foundation. In this method, an auxiliary variable q representing the first-order time derivative of hydrodynamic pressure is treated as the primary unknown for the reservoir domain. Similarly, velocity v is the primary unknown for the solid domain. A three point Gauss–Lobatto quadrature rule is employed for computing time integral of those terms which contain stress term. Further, a partitioned method based on a block-iterative scheme is employed to incorporate the material nonlinearity of the concrete and to enforce the coupling between the two domains in a single iteration loop. A co-axially rotating crack model with exponential strain softening rule is employed to model the fracture of the concrete. Afterwards, numerical performance of the proposed scheme is demonstrated by analyzing two situations of dynamic fracture of Koyna concrete gravity dam.

A space-time finite element method for fractional wave problems

In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order γ (1 < γ < 2). We first establish the stability of the proposed method and then derive the optimal convergence rate in H1(0,T;L2(Ω))-norm and suboptimal rate in discrete $ L^{\infty }(0,T;{H_{0}^{1}}({\Omega })) $ -norm. Furthermore, we discuss the performance of this method when the true solution has singularity at t = 0 and show that optimal convergence rate with respect to H1(0,T;L2(Ω))-norm can still be achieved by using graded temporal grids. Finally, numerical experiments are performed to verify the theoretical results.