Space-time Discontinuous Galerkin Method Research Articles

Numerical analysis of a space-time continuous Galerkin method for the nonlinear Sobolev equation

In this work, we shall use space-time continuous Galerkin (STCG) method to analyze the nonlinear Sobolev equation. This method is different from the conventional finite element methods, that is, it not only uses finite element to discrete spatial variables but also uses finite element to discrete temporal variable. Therefore it is easier to obtain high order accuracy in time and space than the conventional finite element methods and has the good stability. Also, it does not need time step size and space mesh parameter to satisfy the stability conditions. In this paper, we show a detailed analysis including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. Especially, the analysis method given here can be similarly applied to the more general nonlinear problems. At last, the numerical examples reveal that the iterative method is more efficient than the linearized method for the nonlinear problems with STCG method. Also, the numerical experiments show that the STCG method is more efficient than the space-time discontinuous Galerkin (STDG) method in the practical calculations.

Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs

Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs

An Anisotropic hp-mesh Adaptation Method for Time-Dependent Problems Based on Interpolation Error Control

We propose an efficient mesh adaptive method for the numerical solution of time-dependent partial differential equations considered in the fixed space-time cylinder varOmega times (0,T). We employ the space-time discontinuous Galerkin method which enables us to use different meshes at different time levels in a natural way. The mesh adaptive algorithm is based on control of the interpolation error in the L^infty (0,T; L^q(varOmega ))-norm. The goal is to construct a sequence of conforming triangular meshes in such a way that the interpolation error bound is under a given tolerance and the number of degrees of freedom is minimal. The resulting grids consist of anisotropic mesh elements with varying polynomial approximation degrees with respect to space. We present a theoretical framework of this approach as well as several numerical examples demonstrating the accuracy, efficiency, and applicability of the method.

Space-Time Discontinuous Galerkin Methods for Weak Solutions of Hyperbolic Linear Symmetric Friedrichs Systems

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the {textrm{L}}_2 norm at final time. Numerical experiments confirm that the a priori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator.

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step Delta t satisfy hcong CDelta t, with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the L^infty and L^2-norms is order p+1 for polynomial orders p=1 and p=3 and order p for polynomial order p=2.

Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow

This study deals with the development of an accurate, efficient and robust method for the numerical solution of the interaction of compressible flow and nonlinear dynamic elasticity. This problem requires the reliable solution of flow in time-dependent domains and the solution of deformations of elastic bodies formed by several materials with complicated geometry depending on time. In this paper, the fluid–structure interaction (FSI) problem is solved numerically by the space-time discontinuous Galerkin method (STDGM). In the case of compressible flow, we use the compressible Navier–Stokes equations formulated by the arbitrary Lagrangian–Eulerian (ALE) method. The elasticity problem uses the non-stationary formulation of the dynamic system using the St. Venant–Kirchhoff and neo-Hookean models. The STDGM for the nonlinear elasticity is tested on the Hron–Turek benchmark. The main novelty of the study is the numerical simulation of the nonlinear vocal fold vibrations excited by the compressible airflow coming from the trachea to the simplified model of the vocal tract. The computations show that the nonlinear elasticity model of the vocal folds is needed in order to obtain substantially higher accuracy of the computed vocal folds deformation than for the linear elasticity model. Moreover, the numerical simulations showed that the differences between the two considered nonlinear material models are very small.

Superconvergence of the space-time discontinuous Galerkin method for linear nonhomogeneous hyperbolic equations

In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is $$(2n+1)$$ -th order superconvergent at the downwind-biased Radau points in the discrete $$L^2$$ -norm. As a by-product, we obtain a point-wise superconvergence with order $$2n+\frac{1}{2}$$ in vertices. We also find that, in order to obtain these superconvergence results, the source integral term has to be approximated by $$(n+1)$$ -point Radau-quadrature rule. Numerical results are presented to verify our theoretical findings.

NUMERICAL SIMULATION OF A SINGLE RING INFILTRATION EXPERIMENT WITH hp-ADAPTIVE SPACE-TIME DISCONTINUOUS GALERKIN METHOD

We present a novel hp-adaptive space-time discontinuous Galerkin (hp-STDG) method for the numerical solution of the nonstationary Richards equation equipped with Dirichlet, Neumann and seepage face boundary conditions. The hp-STDG method presented in this paper is a generalization of a hp-STDG method which was developed for time dependent non-linear convective-diffusive problems. We describe the method and the single ring experiment, and then we present a numerical experiment which clearly demonstrates the superiority of the hp-STDG method over a discontinuous Galerkin method based on a static fine mesh.

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of degree $N$ and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree $N$ on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, quadrature-free and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efficient Picard iterations. Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time.

A space-time discontinuous Galerkin method for the elastic wave equation

In this work we present a new high order space-time discretization method based on a discontinuous Galerkin paradigm for the second order visco-elastodynamics equation. After introducing the method, we show that the resulting space-time discontinuous Galerkin formulation is well-posed, stable and retains optimal rate of convergence with respect to the discretization parameters, namely the mesh size and the polynomial approximation degree. A set of two and three-dimensional numerical experiments confirms the theoretical bounds.

Statistical Volume Elements for the Characterization of Angle-Dependent Fracture Strengths in Anisotropic Microcracked Materials

Abstract Statistical volume elements (SVEs) are used to homogenize fracture strength of rock, based on the microcrack statistics of a real-world Yuen-Long marble sample. The small size of SVEs enables maintaining inhomogeneities in fracture properties with lower computational cost compared to methods that explicitly model microcracks at macroscale. Maintaining inhomogeneity is important to capture realistic fracture patterns in rock as a quasi-brittle material. Uniaxial tensile, uniaxial compressive, and shear strengths are derived for arbitrary angle for loading and orientation of a single crack by using the linear elastic fracture mechanics (LEFM) method and incorporating frictional effects. Mesoscopic fracture strength fields are generated for different strengths and angle of loading by traversing the spatial domain with circular SVEs. Increasing the SVE size smoothens the spatial inhomogeneity and angular anisotropy of homogenized strengths. Spatial and angular covariance functions of the random fields are obtained to demonstrate how fracture strength varies in space and by changing the angle of loading. Two isotropic and anisotropic rock domains are studied and shown to have very different single- and two-point statistics. Macroscopic fracture simulations by an asynchronous spacetime discontinuous Galerkin (aSDG) method demonstrate that most macroscopic cracks for the anisotropic domain are aligned with the weakest strength planes.

A space-time discontinuous Galerkin method for linear hyperbolic PDE's with high frequencies

The main purpose of this paper is to describe a space-time discontinuous Galerin (DG) method based on an extended space-time approximation space for the linear first order hyperbolic equation that contains a high frequency component. We extend the space-time DG spaces of tensor-product of polynomials by adding trigonometric functions in space and time that capture the oscillatory behavior of the solution. We construct the method by combining the basic framework of the space-time DG method with the extended finite element method. The basic principle of the method is integrating the features of the partial differential equation with the standard space-time spaces in the approximation. We present error analysis of the space-time DG method for the linear first order hyperbolic problems. We show that the new space-time DG approximation has an improvement in the convergence compared to the space-time DG schemes with tensor-product polynomials. Numerical examples verify the theoretical findings and demonstrate the effects of the proposed method.

Statistical volume element averaging scheme for fracture of quasi-brittle materials

To capture the randomness and inhomogeneity of rock at microscale, a statistical volume element (SVE) averaging approach is proposed. The microcrack statistics of a real-world Yuen-Long marble sample is used to realize 2D microcracked domains. The size effect, i.e. the decrease of the mean and variation of homogenized strength field by increasing SVE size, is analyzed. Increasing the crack density is shown to have a similar effect. While smaller SVEs maintain a greater level of inhomogeneity and are preferred for fracture analysis, it is shown that low density of microcracks pose a lower limit on the SVE size. Beside the actual power-law distribution of microcrack length, by varying the Weibull model shape parameter m other domains are created with different microcrack distribution shapes. Macroscopic fracture simulations, by the asynchronous Spacetime Discontinuous Galerkin (aSDG) method, study the effect of m for a uniaxial tensile problem. By increasing m from 0.5 to 4, the length distribution of microcracks become more uniform; this corresponds to a more uniform and stronger mesoscopic strength field, which results to about 3 and 6 times increase to macroscopic tensile strength and toughness, respectively. However, the more uniform length distribution of microcracks is shown to reduce rock brittleness.

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.

Numerical simulation of two-phase flow of immiscible fluids by the finite element, discontinuous Galerkin and level-set methods

The subject of the paper is the numerical simulation of two-phase flow of immiscible fluids. Their motion is described by the incompressible Navier-Stokes equations with different constant density and viscosity for different fluids. The interface between the fluids is defined with the aid of the level-set method using a transport first-order hyperbolic equation. The Navier-Stokes system is equipped with initial and boundary conditions and transmission conditions on the interface between the two fluids. This system is discretized by the Taylor-Hood P2/P1 conforming finite elements in space and the second-order BDF method in time. The transport level-set problem is solved with the aid of the space-time discontinuous Galerkin method. Numerical experiments demonstrate that the developed method is accurate and robust.

Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise

Abstract We consider the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild space–time noise posed on a bounded domain of $\mathbb{R}^2$. The positive parameter $\varepsilon $ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin., 15, 407–438). The noise, although smooth, becomes white on the sharp interface limit $\varepsilon \rightarrow 0^+$. We construct a nonlinear discontinous Galerkin scheme with space–time finite elements of general type that are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer’s Theorem. We first derive abstract error estimates and then, for the case of piecewise polynomial finite elements, we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\varepsilon $. Finally, we present a linear approximation of the nonlinear scheme, for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in $2+1$-dimensional subdomains for a nonlinear parabolic problem. In addition this problem involves noise. These types of schemes avoid any Runge–Kutta-type discretization for the evolutionary variable, and seem to be very effective when applied to equations of such a difficulty.

Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows

This paper is concerned with the numerical simulation of time-dependent variably-saturated Darcian flow problems described by the Richards equation. We present the adaptive higher-order space-time discontinuous Galerkin (hp-STDG) method which optimizes accuracy and efficiency by balancing the errors that arise from the space and time discretizations and from the resulting nonlinear algebraic system. Convergence problems related to the transition between unsaturated flow and saturated flow are eliminated by regularizing the constitutive formulas. We also present an hp-anisotropic mesh adaptation technique capable of generating unstructured triangular elements with optimal sizes, shapes, and polynomial approximation degrees. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the numerical method presented here.

Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of non ...

A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture

We present a stochastic bulk damage model for rock fracture. The decomposition of strain or stress tensor to its negative and positive parts is often used to drive damage and evaluate the effective stress tensor. However, they typically fail to correctly model rock fracture in compression. We propose a damage force model based on the Mohr-Coulomb failure criterion and an effective stress relation that remedy this problem. An evolution equation specifies the rate at which damage tends to its quasi-static limit. The relaxation time of the model introduces an intrinsic length scale for dynamic fracture and addresses the mesh sensitivity problem of earlier damage models. The ordinary differential form of the damage equation makes this remedy quite simple and enables capturing the loading rate sensitivity of strain-stress response. The asynchronous Spacetime Discontinuous Galerkin (aSDG) method is used for macroscopic simulations. To study the effect of rock inhomogeneity, the Karhunen-Loeve method is used to realize random fields for rock cohesion. It is shown that inhomogeneity greatly differentiates fracture patterns from those of a homogeneous rock, including the location of zones with maximum damage. Moreover, as the correlation length of the random field decreases, fracture patterns resemble angled-cracks observed in compressive rock fracture.