Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order $0<α<1$, and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order $1<α<2$. By a new duality technique, pointwise-in-time error estimates of first-order and $ (3-α) $-order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. 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 in [IMA J. Numer. Anal., 33(1) (2013), pp. 242-260] by R. Andreev and in [Comput. Methods Appl. Math., 15(4) (2015), pp. 551-566] by O. Steinbach.
High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes…
Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier--Stokes equations. The spatial discretization based on inf-sup stable pairs of finite element spaces is stabilised using a one-level local projection stabilisation method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for the semi-discrete case. For the fully discrete case, error estimates for both velocity and pressure are given. Numerical results support the theoretical predictions.
We present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.
We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.
A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem
Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations are widely used to solve elliptic equations, and a main ingredient of such solvers is the interpolation operator to transfer information from the coarse to the fine grid. Classical interpolation operators give continuous interpolated values, but since DG solutions are naturally discontinuous, one might wonder if one should not use discontinuous interpolation operators for DG discretizations. We consider a discontinuous interpolation operator with a parameter that controls the discontinuity, and determine the optimal ch…
The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators undiscretized. The well-posedness of this semi-discrete scheme is analyzed and a priori error estimates are derived in the energy norm. We then combine this $hp$-version discontinuous Galerkin method for temporal discretization with an $H^1$-conforming finite element approximation for the spatial variables to construct a fully discrete scheme. A prior error estimates ar…
We present a novel postprocessing technique for a discontinuous Galerkin (DG) discretization of time-dependent Maxwell's equations that we couple with an explicit Runge-Kutta time-marching scheme. The postprocessed electromagnetic field converges one order faster than the unprocessed solution in the H(curl)-norm. The proposed approach is local, in the sense that the enhanced solution is computed independently in each cell of the computational mesh, and at each time step of interest. As a result, it is inexpensive to compute, especially if the region of interest is localized, either in time or …
In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. …
A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations
Centre de Recerca Matemàtica（数学研究中心）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；Department of Mathematics, The Pennsylvania State University（宾夕法尼亚州立大学数学系）
Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, t…
This paper proposes a fully discrete method called the symplectic dG full discretization for stochastic Maxwell equations driven by additive noises, based on a stochastic symplectic method in time and a discontinuous Galerkin (dG) method with the upwind fluxes in space. A priori $H^k$-regularity ($k\in\{1,2\}$) estimates for the solution of stochastic Maxwell equations are presented, which have not been reported before to the best of our knowledge. These $H^k$-regularities are vital to make the assumptions of the mean-square convergence analysis on the initial fields, the noise and the medium …
In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably inv…
A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different discretization variables, $u_h, \bm{p}_h, \check u_h$ and $\check p_h$, where $u_h$ and $\bm{p}_h$ are for approximation of $u$ and $\bm{p}=-α\nabla u$ inside each element, and $ \check u_h$ and $\check p_h$ are for approximation of residual of $u$ and $\bm{p} \cdot \bm{n}$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy in…
Evaluating High Order Discontinuous Galerkin Discretization of the Boltzmann Collision Integral in $O(N^2)$ Operations Using the Discrete Fourier Transform
Department of Mathematics, California State University Nortrhidge, Nortrhdige, CA 91330, USA
We propose and study numerically the implicit approximation in time of the Navier-Stokes equations by a Galerkin-collocation method in time combined with inf-sup stable finite element methods in space. The conceptual basis of the Galerkin-collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is ensured furthe…
We propose a hybrid spatial discretization for the radiative transport equation that combines a second-order discontinuous Galerkin (DG) method and a second-order finite volume (FV) method. The strategy relies on a simple operator splitting that has been used previously to combine different angular discretizations. Unlike standard FV methods with upwind fluxes, the hybrid approach is able to accurately simulate problems in scattering dominated regimes. However, it requires less memory and yields a faster computational time than a uniform DG discretization. In addition, the underlying splitting…
For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple const…
A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems
Massachusetts Institute of Technology（麻省理工学院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；State University of New York Polytechnic Institute（纽约州立理工大学）；Worcester Polytechnic Institute（伍斯特理工学院）
The paper describes the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. The construction takes advantage of existing theory for entropy-stable summation-by-parts (SBP) discretizations. In particular, the paper shows how DGD discretizations -- both linear and nonlinear -- can be constructed by defining the SBP trial and test functions in terms of interpolated DGD degrees of freedom. In the case of entropy-stable discretizations, the entropy variables rather than the conservative variables must be inte…
Isogeometric Analysis is a spline-based discretization method to partial differential equations which shows the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of Isogeometric Analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be …
