DG方法求解对流扩散方程的超收敛和一致收敛性
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摘要
1973年Reed和Hill[72]在求解中子输运方程(一阶双曲型方程)时提出了间断有限元方法(the discontinuous Galerkin finite element method,简记为DG)。自此之后,用间断有限元求解双曲问题和椭圆问题的研究几乎同时得到了快速的发展。1997年,Bassi和Rebay[11]设计了一种间断有限元方法求解Navier-Stokes方程,并且获得了稳定的高阶收敛格式。受Bassi和Rebay数值实验结果的启发,Cockburn和Shu[36]提出了局部间断有限元法(简记为LDG),同时Baumann和Oden[8]也发展了一种新的DG方法。现在DG方法已经被广泛应用于求解双曲守恒律组、椭圆方程、对流扩散方程、Hamilton-Jacobia方程,以及KdV方程等。关于DG方法的全面综述及其应用参看[31]。
用间断有限元方法求解对流扩散问题是近年来的热门研究课题。受DG方法求解双曲型方程巨大成功的启发,本文用DG方法求解对流扩散方程。证明了一大类间断有限元方法求解定常对流扩散问题的存在唯一性,并且得到了u和q=u'的离散误差的一个渐进展开表达式。对md-LDG方法,间断有限元解U和其导数Q的离散误差的主项分别与每个单元的p+1阶右Radau和左Radau多项式成比例。事实上,单元内部的p+1阶右Radau点和左Radau点分别是U和Q的p+2阶超收敛点。对其它满足相容性和守恒性的DG方法,在一定的假设条件下,其间断有限元解U和其导数Q的离散误差的主项都与p阶Legendre多项式成比例。因此间断有限元解的导数Q在Gauss点有p+1阶超收敛。基于这些超收敛性质,我们得到了一个后验误差估计。数值例子证实了理论证明的可靠性。
当扩散系数ε趋向于零时,一般的数值方法在均匀网格下求解奇异摄动对流扩散问题,不能得到一致收敛的格式。本文在两种局部加密网格(即Shishkin网格和改进的等级网格)下,用LDG方法求解一维和二维奇异摄动对流扩散问题。数值结果表明,对任意小的ε,即使在均匀网格下,对一维和二维情形,LDG方法都不会产生任何非物理震荡。在Shishkin网格和改进的等级网格下,数值通量有2p+1阶一致超收敛。改进的等级网格不但保持了Shishkin网格原有的优点,而且更有效更稳定。值得指出的是,一致收敛性的理论证明非常困难,有待进一步研究。
最后本文设计了一种新的DG方法求解对流扩散方程,随后对解的存在唯一性给出了证明。在节点处数值流通量有2p+1阶超收敛,在改进的等级网格下,DG解还具有一致的收敛性。
The discontinuous Galerkin method(DGM) was first introduced for the neutron transport equation(The first-order hyperbolic equation) in 1973 by Reed and Hill[72].Since then there has been an active development of DG methods for hyperbolic and elliptic equations in parallel.In 1997,Bassi and Rebay[11]provided a discontinuous Galerkin method for solving compressible Navier-Stokes equations,and obtained a stable and high-order convergent scheme.Motivated by the successful numerical experiments of Bassi and Rebay, Cockburn and Shu[36]developed the local discontinuous Galerkin method (LDG).At the same time,a discontinuous Galerkin method was introduced by Baumann and Oden[8].Now the DG methods have been used widely for hyperbolic conservation laws systems,elliptic equations,convection-diffusion equations,Hamilton-Jacobi equations and KdV equations,etc.For a fairly thorough compilation of the history of these methods and their applications see[31].
In recent years,the DG methods for convection-diffusion problems have been one of the highlights in the study of numerical methods.Inspired by the great success of the DG method in solving hyperbolic equations,in this paper the discontinuous Galerkin method for convection-diffusion equation would be studied.The existence and uniqueness of the approximate solutions for a class of DG methods are proved.Then we will present the asymptotic expansions of the discretization errors for both the potential u and its derivative q = u' for a class of DG methods.For md-LDG method,the leading terms for the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p + 1 on each element,respectively.This fact implies that the p + 1 degree right Radau points and left Radau points are the p + 2 degree superconvergence points for U and Q,separately.Nevertheless, for other DG methods which are consistent and conservative and satisfy some assumptions,the leading terms for the errors of U and Q are proportional to Legendre polynomial of degree p,respectively.As a result,only the p+1-degree Gauss points are the p + 1-order superconvergence points for Q.A Posteriori Error estimates based on superconvergence property will be developed.Our numerical experiment verify our theoretical results.
On the other hand,whenεis small,under the uniform mesh,the classical numerical methods cannot produce a scheme of uniform convergence.In this paper we will compare two-type layer-adapted meshes,i.e.,Shishkin mesh and improved grade meshes,when they are used in the h-version of the LDG method for one and two dimensional problems.The numerical results exhibit that the LDG method does not produce any oscillation even under uniform meshes for arbitraryεfor both 1-D and 2-D cases.On the other hand,the 2p + 1-order uniform superconvergence of numerical fluxes are observed numerically for the LDG method under both the Shishkin and improved grade meshes.The numerical results indicate that the improved grade meshes not only keep the advantages of the Shishkin meshes,but is also more efficient and stable than the Shishkin meshes.It is worthwhile to point out that theoretical analysis of the uniform convergence is extremely difficult and remains an open problem for the LDG method
A robust DG scheme will be designed to solve the singularly perturbed convection-diffusion equations in one-dimensional setting with Dirichlet boundary conditions.The existence and uniqueness of approximate solutions are proved.The 2p + 1-order superconvergence of numerical traces at each node is observed.We also note that this DG method is robust with respect to the diffusion coefficientεunder the improved grade mesh.
用间断有限元方法求解对流扩散问题是近年来的热门研究课题。受DG方法求解双曲型方程巨大成功的启发,本文用DG方法求解对流扩散方程。证明了一大类间断有限元方法求解定常对流扩散问题的存在唯一性,并且得到了u和q=u'的离散误差的一个渐进展开表达式。对md-LDG方法,间断有限元解U和其导数Q的离散误差的主项分别与每个单元的p+1阶右Radau和左Radau多项式成比例。事实上,单元内部的p+1阶右Radau点和左Radau点分别是U和Q的p+2阶超收敛点。对其它满足相容性和守恒性的DG方法,在一定的假设条件下,其间断有限元解U和其导数Q的离散误差的主项都与p阶Legendre多项式成比例。因此间断有限元解的导数Q在Gauss点有p+1阶超收敛。基于这些超收敛性质,我们得到了一个后验误差估计。数值例子证实了理论证明的可靠性。
当扩散系数ε趋向于零时,一般的数值方法在均匀网格下求解奇异摄动对流扩散问题,不能得到一致收敛的格式。本文在两种局部加密网格(即Shishkin网格和改进的等级网格)下,用LDG方法求解一维和二维奇异摄动对流扩散问题。数值结果表明,对任意小的ε,即使在均匀网格下,对一维和二维情形,LDG方法都不会产生任何非物理震荡。在Shishkin网格和改进的等级网格下,数值通量有2p+1阶一致超收敛。改进的等级网格不但保持了Shishkin网格原有的优点,而且更有效更稳定。值得指出的是,一致收敛性的理论证明非常困难,有待进一步研究。
最后本文设计了一种新的DG方法求解对流扩散方程,随后对解的存在唯一性给出了证明。在节点处数值流通量有2p+1阶超收敛,在改进的等级网格下,DG解还具有一致的收敛性。
The discontinuous Galerkin method(DGM) was first introduced for the neutron transport equation(The first-order hyperbolic equation) in 1973 by Reed and Hill[72].Since then there has been an active development of DG methods for hyperbolic and elliptic equations in parallel.In 1997,Bassi and Rebay[11]provided a discontinuous Galerkin method for solving compressible Navier-Stokes equations,and obtained a stable and high-order convergent scheme.Motivated by the successful numerical experiments of Bassi and Rebay, Cockburn and Shu[36]developed the local discontinuous Galerkin method (LDG).At the same time,a discontinuous Galerkin method was introduced by Baumann and Oden[8].Now the DG methods have been used widely for hyperbolic conservation laws systems,elliptic equations,convection-diffusion equations,Hamilton-Jacobi equations and KdV equations,etc.For a fairly thorough compilation of the history of these methods and their applications see[31].
In recent years,the DG methods for convection-diffusion problems have been one of the highlights in the study of numerical methods.Inspired by the great success of the DG method in solving hyperbolic equations,in this paper the discontinuous Galerkin method for convection-diffusion equation would be studied.The existence and uniqueness of the approximate solutions for a class of DG methods are proved.Then we will present the asymptotic expansions of the discretization errors for both the potential u and its derivative q = u' for a class of DG methods.For md-LDG method,the leading terms for the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p + 1 on each element,respectively.This fact implies that the p + 1 degree right Radau points and left Radau points are the p + 2 degree superconvergence points for U and Q,separately.Nevertheless, for other DG methods which are consistent and conservative and satisfy some assumptions,the leading terms for the errors of U and Q are proportional to Legendre polynomial of degree p,respectively.As a result,only the p+1-degree Gauss points are the p + 1-order superconvergence points for Q.A Posteriori Error estimates based on superconvergence property will be developed.Our numerical experiment verify our theoretical results.
On the other hand,whenεis small,under the uniform mesh,the classical numerical methods cannot produce a scheme of uniform convergence.In this paper we will compare two-type layer-adapted meshes,i.e.,Shishkin mesh and improved grade meshes,when they are used in the h-version of the LDG method for one and two dimensional problems.The numerical results exhibit that the LDG method does not produce any oscillation even under uniform meshes for arbitraryεfor both 1-D and 2-D cases.On the other hand,the 2p + 1-order uniform superconvergence of numerical fluxes are observed numerically for the LDG method under both the Shishkin and improved grade meshes.The numerical results indicate that the improved grade meshes not only keep the advantages of the Shishkin meshes,but is also more efficient and stable than the Shishkin meshes.It is worthwhile to point out that theoretical analysis of the uniform convergence is extremely difficult and remains an open problem for the LDG method
A robust DG scheme will be designed to solve the singularly perturbed convection-diffusion equations in one-dimensional setting with Dirichlet boundary conditions.The existence and uniqueness of approximate solutions are proved.The 2p + 1-order superconvergence of numerical traces at each node is observed.We also note that this DG method is robust with respect to the diffusion coefficientεunder the improved grade mesh.
引文
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