共查询到20条相似文献,搜索用时 31 毫秒
1.
Lunji Song 《Numerical Methods for Partial Differential Equations》2012,28(1):288-311
In this article, we investigate interior penalty discontinuous Galerkin (IPDG) methods for solving a class of two‐dimensional nonlinear parabolic equations. For semi‐discrete IPDG schemes on a quasi‐uniform family of meshes, we obtain a priori bounds on solutions measured in the L2 norm and in the broken Sobolev norm. The fully discrete IPDG schemes considered are based on the approximation by forward Euler difference in time and broken Sobolev space. Under a restriction related to the mesh size and time step, an hp ‐version of an a priori l∞(L2) and l2(H1) error estimate is derived and numerical experiments are presented.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 288–311, 2012 相似文献
2.
Mi Ray Ohm Hyun Young Lee Jun Yong Shin 《Journal of Applied Mathematics and Computing》2011,37(1-2):473-495
In this paper we develop fully discrete discontinuous Galerkin approximation using symmetric interior penalty method. We construct finite element spaces which consist of piecewise continuous polynomials. We introduce an appropriate elliptic-type projection of u and prove its optimal convergence. We develop fully discrete discontinuous Galerkin approximations and prove the optimal convergence in ? ∞(L 2) normed space. 相似文献
3.
Clint Dawson Jennifer Proft 《Numerical Methods for Partial Differential Equations》2001,17(6):545-564
The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L∞(L2) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L2(L2) norm for the first penalty method with a penalty parameter of order one (h?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001 相似文献
4.
Kaixin Wang Hong Wang Mohamed Al‐Lawatia 《Numerical Methods for Partial Differential Equations》2010,26(3):561-595
We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L2 norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 相似文献
5.
Václav Kučera 《Numerical Functional Analysis & Optimization》2013,34(3):285-312
This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection–diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L ∞(L 2)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular. 相似文献
6.
A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi‐discrete and a family of fully‐discrete time approximate schemes are formulated. These schemes are symmetric. Hp‐version error estimates are analyzed for these schemes. For the semi‐discrete time scheme a priori L∞(H1) error estimate is derived and similarly, l∞(H1) and l2(H1) for the fully‐discrete time schemes. These results indicate that spatial rates in H1 and time truncation errors in L2 are optimal. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
7.
Miloslav Feistauer Václav Kučera Karel Najzar Jaroslava Prokopová 《Numerische Mathematik》2011,117(2):251-288
The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary
convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly
perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in
general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary
penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates
in “L
2(L
2)”- and “DG”-norm formed by the “L
2(H
1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration
has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect
to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour
in time as a polynomial of degree ≤ q. 相似文献
8.
Shimin Chai Yongkui Zou Chenguang Zhou Wenju Zhao 《Numerical Methods for Partial Differential Equations》2019,35(5):1745-1755
This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L2 and an H2 type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results. 相似文献
9.
M. Vlasák V. Dolejší J. Hájek 《Numerical Methods for Partial Differential Equations》2011,27(6):1456-1482
We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L∞(L2) –norm and the L2(H1) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010 相似文献
10.
Chaobao Huang Martin Stynes 《Numerical Methods for Partial Differential Equations》2019,35(6):2076-2090
A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L2(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L2(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H1(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp. 相似文献
11.
Strong convergence of the discontinuous Galerkin scheme for the low regularity miscible displacement equations 下载免费PDF全文
Vivette Girault Jizhou Li Beatrice M. Rivière 《Numerical Methods for Partial Differential Equations》2017,33(2):489-513
Strong convergence of the numerical solution to a weak solution is proved for a nonlinear coupled flow and transport problem arising in porous media. The method combines a mixed finite element method for the pressure and velocity with an interior penalty discontinuous Galerkin method in space for the concentration. Using functional tools specific to broken Sobolev spaces, the convergence of the broken gradient of the numerical concentration to the weak solution is obtained in the L2 norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 489–513, 2017 相似文献
12.
The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear
nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately
in space and time using, in general, different nonconforming space grids on different time levels and different polynomial
degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty
approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L
2(L
2)”-and “
”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization,
we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution,
the errors are of order O(h
p
+ τ
q
). The estimates hold true even in the hyperbolic case when ɛ = 0. 相似文献
13.
Jiansong
Zhang Huiran Han Hui Guo Xiaomang Shen 《Numerical Methods for Partial Differential Equations》2020,36(6):1629-1647
In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L∞(L2) for velocity and concentration and the super convergence in L∞(H1) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis. 相似文献
14.
Tianliang Hou 《Applicable analysis》2013,92(8):1655-1665
In this article, we analyse a posteriori error estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ?≥?1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r?≥?1). Using the duality argument, we derive a posteriori l ∞(L 2) error estimates for the scalar function, assuming that only the underlying mesh is static. 相似文献
15.
Chen Liu Florian Frank Batrice M. Rivire 《Numerical Methods for Partial Differential Equations》2019,35(4):1509-1537
In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn–Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori error estimates. Our analysis is valid for both symmetric and nonsymmetric versions of the discontinuous Galerkin formulation. 相似文献
16.
A compact C0 discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C0 finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C0 interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H1‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015 相似文献
17.
Zhaojie Zhou Xiaoming Yu Ningning Yan 《Numerical Methods for Partial Differential Equations》2014,30(1):339-360
In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L2 norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014 相似文献
18.
A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated.A symmetric interior penalty discontinuous Galerkin (SIPG) method is applied to the coupled system of flow and transport.Using the induction hypotheses instead of the cut-off operator and the interpolation projection properties,a priori hp error estimates are presented.The error bounds in L2(H1) norm for concentration and in L∞(L2) norm for velocity are optimal in h and suboptimal in p with a loss of power 1/2. 相似文献
19.
Meriem Zefzouf Fabien Marche 《Numerical Methods for Partial Differential Equations》2023,39(2):1478-1503
In this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one-dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non-conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well-posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments. 相似文献
20.
Bhupen Deka 《Numerical Methods for Partial Differential Equations》2019,35(5):1630-1653
We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L∞(L2) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L∞(L2) norm. 相似文献