四边形网上有限体积元法的L~2误差估计及超收敛
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摘要
本文以二阶椭圆型方程为模型,对求解区域作凸四边形剖分,在试探函数空间分别取等参双线性有限元空间及等参双二次有限元空间两种情形下,研究了相应的有限体积元法的最佳阶L2误差估计及超收敛性.
对于等参双线性元有限体积法,本文在h2拟平行四边形条件下,给出了最佳阶的L2误差估计.其主要思想是在对偶论证法的框架下,将证明归结为估计有限元与有限体积两者双线性形式的差.我们将其分解成两部分:原始单元上的积分求和以及原始单元边界上的线积分求和.对于第一部分,我们精确计算了某些积分,在h2拟平行四边形条件下,得到了所要的估计;对于第二部分,除了要精确计算积分外,还要将这些线积分按单元对边合并,进而在h2拟平行四边形条件下,得到了所需的估计.接着,借助于单元间的合并技巧以及Bramble-Hilbert引理,在h2均匀四边形网上,我们证明了双线性元有限体积法的第一插值弱估计为2阶.进而得出,在三类插值超收敛点上,导数逼近按平均模意义具有超收敛.
对于等参双二次元有限体积法,已有的文献均只构造了相应的格式并证明了数值解按H1模2阶收敛.数值试验表明这些格式的L2模收敛阶均仅为2,而且在单元内部的插值应力佳点上,导数的超收敛性也不存在.本文重新选取了对偶剖分方式取单元边界线上的2阶Gauss点作为对偶剖分节点,取单元内部Gauss线作为对偶单元边界线构造了新的有限体积元格式.本文在h2拟平行四边形条件下,证明了新格式按H1模2阶收敛.同时,我们用一种新方法证得了此格式的最佳阶L2误差估计.与上面双线性元有限体积法的L2估计证明方法不同,这种新方法是将有限体积法的双线性形式按原始单元求和,把限制在原始单元的部分,即有限体积法的单元二次型,看成是有限元法单元二次型的某种数值积分公式.通过估计求积余项,得到有限体积与有限元两者双线性形式的差的估计,进而得到最佳的L2估计.而且,利用这种新方法,我们成功地证明了双线性元有限体积法的最佳阶L2估计.同时,对于矩形网上双二次元有限体积法,我们证明了它的第一插值弱估计为3阶,进而得到在单元内部2阶Gauss点(即插值应力佳点)上,有限体积解的导数具有超收敛性.
本文在矩形网及几种四边形网上做了详细的数值实验,验证了等参双线性元的最佳阶L2估计及超收敛性,等参双二次元的最佳阶L2估计及超收敛.虽然我们仅在矩形网上证明了等参双二次元的超收敛,但数值实验表明,这种超收敛性可以推广到四边形网上.
The Finite Volume Element Method, also called as Generalized Di?erence Method,was firstly put up by professor Li Ronghua in 1982. Its computational simplicity andpreserving local conservation of certain physical quantities,make it be widely used incomputing ?uid mechanics and electromagnetic field and other fields.
In this paper, we consid?er the following Poisson equation:
where ? ? R2 is a bounded convex polygon with boundaryΓ= ??, and f∈L2(?).The corresponding variational problem for (0-1) is: find u∈U := H01(?) satisfyingwhere
Divide ? into a convex quadrilateral partition Th. Suppose that Th is quasi-uniform, i.e., there exist positive constants C1 ,C2 such that
Suppose that the following regular condition be satisfied for any quadrilateral K∈Th:
(i) There exist constants C1,C2 > 0 independent of h, such that
(ii) For any interior angleθof K∈Th, there exist constantsθ0 > 0 independentof h, such that
We take the unit square K = [0,1]×[0,1] on the (ξ,η) plane as a reference element.For any convex quadrilateral K, there exist a unique invertible bilinear transformationFK which maps K onto K.
Under the above conditions, we obtain the following results:
I. The Optimal L2 Error Estimates For The Isoparametric Bilinear FiniteVolume Element Methods
The trial function space Uh is taken as isoparametric bilinear finite element spaceon quadrilateral partition, and the test function space Vh is defined as piecewise con-stant space on dual partition. So, the finite volume method for (0-1) is : find uh∈Uhsuch thatwhere
We call that quadrilateral element K = P1P2P3P4 satisfies quasi-parallel quadrilat-eral condition, if
In fact, we give two methods to prove the optimal L2 error estimats.
i. The First Method
Introduce an auxiliary problem: for ?g∈L2(?), find w∈H01(?) such that
We assume that (0-2) and (0-8) are regular, i.e., for any f,g∈L2(?), the correspondingsolution u = uf and w = wg are in the space H01(?)∩H2(?), and there exists a constantC such that
Lemma 0.1 It holds that(g,u?uh) = a(u?uh,w?Πhw)+(f,Πhw?Π?hw)+ah(uh,Πh?w)?a(uh,Πhw). (0-10)
Noticing that |a(u ? uh,w ?Πhw)|≤Ch2||u||2||w||2≤Ch2||f||0||g||0, we needonly to estimate |(f,Πhw ?Πh?w)| and |ah(uh,Πh?w) ? a(uh,Πhw)|.
Lemma 0.2 Suppose that (0-3), (0-4), (0-5) and (0-7) are satisfied, and f∈H1(?), then there exist constant C independent of h such that
Write ah(uh,Πh?w) ? a(uh,Πhw) = E1 + E2, where
In the sequel, we estimate E1 and E2 element by element. We have E1 = 0when K is a parallelogram, because ??2xu2h and ??2yu2h are both constants, and K(Πhw ?Πh?w)dxdy = 0. And for E2, we calculate the integrals on the boundaries of everyelement exactly, and combine the integrals on opposite edges and estimate them. Withthe quasi-parallel quadrilateral condition (0-7), we obtain |E1|≤Ch2||f||0||g||0,|E2|≤Ch2||f||0||g||0, respectively. Moreover, we conjecture that the su?cient condition (0-7)can not be weakened any more.
Lemma 0.3 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5) and(0-7) are satisfied, and f∈L2(?), then
Lemma 0.4 Under assumptions of Lemma 0.3, we have
With the above four lemmas, we have the following L2 estimate:
Theorem 0.1 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied, and let u∈H01(?)∩H2(?) be the solution to (0-1), anduh∈Uh to (0-6), then for any f∈H1(?), we have
ii. The Second Method
Notice thatwe need only to estimate |a(u ? uh,Πhw) ? ah(u ? uh,Πh?w)|.
Theorem 0.2 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied, and let u∈H01(?)∩H3(?) be the solution to (0-1), anduh∈Uh to (0-6), then we have
II. The Superconvergence For The Isoparametric Bilinear Finite VolumeElement Methods
As we know, in the error analysis, the determination of the upper bound of ||u ?uh||m is usually reduced to the estimation of ||u ?Πhu||m and a(u ?Πhu,Πh?wh). Bythe approximation theory, in general we can only obtain, limited by degree k of theapproximate polynomials, that
In general this estimate can not be improved even if the solution u possesses an highersmoothness. Therefore,
is the optimal order error estimate. But this face does not exclude the possibility thatthe approximation of the derivatives may be of higher order accuracy at some specialpoints, called optimal stress points.
Just as in the case of finite element methods, we can also obtain superconvergenceresults for finite volume methods, provided we manage to get the super interpolationweak estimates. In detail we have the following theorem.
Lemma 0.5 Let u and uh be solutions of the boundary value problem andits finite volume element scheme, respectively. Assume the bilinear form of the finitevolume element scheme satisfies the following interpolation weak estimate: There existsp∈[1,∞] such that then one has
Moreover, let Nk be the set of interpolation optimal stress points: For any P∈Nk,there exists q∈[1,+∞] such thatThen one has ?where r is the number of points in Nk.
To get the superconvergence, the partition Th has to satisfy the h2 uniform con-dition, which also be used in the finite element methods.
Call that the quadrilateral partition Th is h2 uniform, if any two h2 quasi-parallelquadrilateral elements K1 = P1P2P3P4, K2 = P4P3P5P6, which share the commonedge P3P4, satisfy
Under the h2 uniform condition, we prove the first weak estimate of interpola-tion to 2 order, further more, we get the superconvergence of the solutions of thefinite volume element methods at three type points, at which the interpolation possessuperconvergence under the average gradient norm.
Theorem 0.3 Suppose that Th is a h2 uniform quadrilateral partition, let u∈H01(?)∩H3(?)∩W2,∞(?), then
Theorem 0.4 Suppose that Th is a h2 uniform quadrilateral partition, let u∈H01(?)∩H3(?)∩W2,∞(?) be the solution to (0-1), uh∈Uh to (0-6), then
consequently
where Si, i = 1,2,3 denote three types of set of optimal of stresses for Uh-interpolation,i.e. the centers of the elements, the inner vertices and the midpoints of inner edges,and denotes the average value of gradient , ri is the number of points in Si.
III. New Formula And Convergence Analysis Of Isoparametric Bi-quadratic Finite Volume Element Methods
For the Lagrange type finite element space, besides the isoparametric bilinearelements, the isoparametric biquadratic elements are also used commonly. The quarterpoints and three-quarter points of edges of one quadrilateral are taken as the nodes ofthe dual partition in existing formula. It seems reasonable to choose this, because heformula guarantee 2 order convergence in H1 norm. However, they are not the optimalscheme, since the convergence order is only 2 in L2 norm. We choose a new plan toget dual partition, construct a new FVEM formula, and prove the optimal H1 and L2error estimates. Moreover, we present detailed numerical experiments.
For any K∈Th, with vertices Pi,i = 1,2,3,4, taking points Pij, such that
Then the control volumes in K of a vertex Pi, a midpoint Mi and averaging center Qare the subregions PiPi(i+1)PiPi(i?1), Pi(i+1)P(i+1)iPi+1Pi and P1P2P3P4 respectively.
Then for each vertex P, we associate a control volume KP?, which is built by theunion of the above subregions, sharing the vertex P. For each midpoint M on anelement edge, we associate a control volume KM?, which is built by the union of theabove subregions, sharing a common edge including M. The set of all the controlvolumes makes the dual partition Th ? .
The trial function space Uh is taken as isoparametric biquadratic finite elementspace on quadrilateral partition, and the test function space Vh is defined as piecewiseconstant space on dual partition. The finite volume method for (0-1) is : find uh∈Uhsuch that where
Firstly, we obtain the coercivity of the bilinear form.
Lemma 0.6 Suppose that every element in Th satisfies the regular conditionand h2 quasi-parallel quadrilateral condition. Then for su?ciently small h, there exista constant C0 > 0 such that
Further more, we prove the H1 error estimate.
Theorem 0.5 Suppose that u∈H01(?)∩H3(?) is the solution to equation (0-1), uh is the solution to the finite volume formula (y:2:1.2.9), If conditions (0-4), (0-5)and (0-7) hold, then there exist a positive constant C independent of h such that
We also get the optimal L2 error estimates.
Theorem 0.6 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied. Let u∈H01(?)∩H4(?) and uh∈Uh are the solutions to (0-1)and(0-6) respectively, then
IV. Superconvergence Analysis Of Isoparametric Biquadratic Finite Vol-ume Element Methods on rectangular meshes
As we know, the four Gauss points in every quadrilateral are the optimal pointsof stresses for isoparametric biquadratic Uh-interpolation. the solutions of the finiteelement methods are superconvergent at these optimal points of stresses. In this paper,we prove that the derivative of the solutions for the isoparametric biquadratic FVEMare also superconvergent at these points. Numerical experiments show that the super-convergence also holds on quadrilateral meshes. But we can’t prove it with the methodused on rectangular meshes.
Firstly, we introduce Legendre orthogonal polynomials on E = (?1,1)
When i = j, we have (li,lj) = 0, and (lj,lj) = 2j2+1, lj(±1) = (±1)j. We obtain theM type polynomials from the integral on lj
Obviously, if n 2, Mn(±1) = 0. And if j ? i = 0 or±2, (Mi,Mj) = 0, (Mi,Mj) = 0otherwise.
To study the superconvergence of the finite volume methods for 2nd order ellipticequations, we will construct the M type expansion polynomials on square K = [?1,1]×[?1,1]. Suppose that u is smooth enough. Fix t, expand u on s to obtainthen for every bi(t), we take M type expand on twith coe?cients
So, we can obtain the M type expansion for u with these coe?cients
Lemma 0.7 Suppose u∈W4,p(?), 1 p∞, then we have the following estimates of the interpolation uIwhere, Z is the set of the Gauss points in element.
Theorem 0.7 Suppose that Th is a regular rectangular partition, let u∈H01(?)∩H4(?), and uI be the orthogonal interpolation polynomial of u, then
We have the following superclose estimate:
Theorem 0.8 Let Th be a regular rectangular partition, If u∈H01(?)∩H4(?)and uh∈Uh are the solutions to (0-1) and (0-18) respectively, thenfurther
Where S is the set of the optimal points of stresses for Uh interpolation, i.e. the Gausspoints in element. denotes the average value of gradient , r is the number of pointsin S.
对于等参双线性元有限体积法,本文在h2拟平行四边形条件下,给出了最佳阶的L2误差估计.其主要思想是在对偶论证法的框架下,将证明归结为估计有限元与有限体积两者双线性形式的差.我们将其分解成两部分:原始单元上的积分求和以及原始单元边界上的线积分求和.对于第一部分,我们精确计算了某些积分,在h2拟平行四边形条件下,得到了所要的估计;对于第二部分,除了要精确计算积分外,还要将这些线积分按单元对边合并,进而在h2拟平行四边形条件下,得到了所需的估计.接着,借助于单元间的合并技巧以及Bramble-Hilbert引理,在h2均匀四边形网上,我们证明了双线性元有限体积法的第一插值弱估计为2阶.进而得出,在三类插值超收敛点上,导数逼近按平均模意义具有超收敛.
对于等参双二次元有限体积法,已有的文献均只构造了相应的格式并证明了数值解按H1模2阶收敛.数值试验表明这些格式的L2模收敛阶均仅为2,而且在单元内部的插值应力佳点上,导数的超收敛性也不存在.本文重新选取了对偶剖分方式取单元边界线上的2阶Gauss点作为对偶剖分节点,取单元内部Gauss线作为对偶单元边界线构造了新的有限体积元格式.本文在h2拟平行四边形条件下,证明了新格式按H1模2阶收敛.同时,我们用一种新方法证得了此格式的最佳阶L2误差估计.与上面双线性元有限体积法的L2估计证明方法不同,这种新方法是将有限体积法的双线性形式按原始单元求和,把限制在原始单元的部分,即有限体积法的单元二次型,看成是有限元法单元二次型的某种数值积分公式.通过估计求积余项,得到有限体积与有限元两者双线性形式的差的估计,进而得到最佳的L2估计.而且,利用这种新方法,我们成功地证明了双线性元有限体积法的最佳阶L2估计.同时,对于矩形网上双二次元有限体积法,我们证明了它的第一插值弱估计为3阶,进而得到在单元内部2阶Gauss点(即插值应力佳点)上,有限体积解的导数具有超收敛性.
本文在矩形网及几种四边形网上做了详细的数值实验,验证了等参双线性元的最佳阶L2估计及超收敛性,等参双二次元的最佳阶L2估计及超收敛.虽然我们仅在矩形网上证明了等参双二次元的超收敛,但数值实验表明,这种超收敛性可以推广到四边形网上.
The Finite Volume Element Method, also called as Generalized Di?erence Method,was firstly put up by professor Li Ronghua in 1982. Its computational simplicity andpreserving local conservation of certain physical quantities,make it be widely used incomputing ?uid mechanics and electromagnetic field and other fields.
In this paper, we consid?er the following Poisson equation:
where ? ? R2 is a bounded convex polygon with boundaryΓ= ??, and f∈L2(?).The corresponding variational problem for (0-1) is: find u∈U := H01(?) satisfyingwhere
Divide ? into a convex quadrilateral partition Th. Suppose that Th is quasi-uniform, i.e., there exist positive constants C1 ,C2 such that
Suppose that the following regular condition be satisfied for any quadrilateral K∈Th:
(i) There exist constants C1,C2 > 0 independent of h, such that
(ii) For any interior angleθof K∈Th, there exist constantsθ0 > 0 independentof h, such that
We take the unit square K = [0,1]×[0,1] on the (ξ,η) plane as a reference element.For any convex quadrilateral K, there exist a unique invertible bilinear transformationFK which maps K onto K.
Under the above conditions, we obtain the following results:
I. The Optimal L2 Error Estimates For The Isoparametric Bilinear FiniteVolume Element Methods
The trial function space Uh is taken as isoparametric bilinear finite element spaceon quadrilateral partition, and the test function space Vh is defined as piecewise con-stant space on dual partition. So, the finite volume method for (0-1) is : find uh∈Uhsuch thatwhere
We call that quadrilateral element K = P1P2P3P4 satisfies quasi-parallel quadrilat-eral condition, if
In fact, we give two methods to prove the optimal L2 error estimats.
i. The First Method
Introduce an auxiliary problem: for ?g∈L2(?), find w∈H01(?) such that
We assume that (0-2) and (0-8) are regular, i.e., for any f,g∈L2(?), the correspondingsolution u = uf and w = wg are in the space H01(?)∩H2(?), and there exists a constantC such that
Lemma 0.1 It holds that(g,u?uh) = a(u?uh,w?Πhw)+(f,Πhw?Π?hw)+ah(uh,Πh?w)?a(uh,Πhw). (0-10)
Noticing that |a(u ? uh,w ?Πhw)|≤Ch2||u||2||w||2≤Ch2||f||0||g||0, we needonly to estimate |(f,Πhw ?Πh?w)| and |ah(uh,Πh?w) ? a(uh,Πhw)|.
Lemma 0.2 Suppose that (0-3), (0-4), (0-5) and (0-7) are satisfied, and f∈H1(?), then there exist constant C independent of h such that
Write ah(uh,Πh?w) ? a(uh,Πhw) = E1 + E2, where
In the sequel, we estimate E1 and E2 element by element. We have E1 = 0when K is a parallelogram, because ??2xu2h and ??2yu2h are both constants, and K(Πhw ?Πh?w)dxdy = 0. And for E2, we calculate the integrals on the boundaries of everyelement exactly, and combine the integrals on opposite edges and estimate them. Withthe quasi-parallel quadrilateral condition (0-7), we obtain |E1|≤Ch2||f||0||g||0,|E2|≤Ch2||f||0||g||0, respectively. Moreover, we conjecture that the su?cient condition (0-7)can not be weakened any more.
Lemma 0.3 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5) and(0-7) are satisfied, and f∈L2(?), then
Lemma 0.4 Under assumptions of Lemma 0.3, we have
With the above four lemmas, we have the following L2 estimate:
Theorem 0.1 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied, and let u∈H01(?)∩H2(?) be the solution to (0-1), anduh∈Uh to (0-6), then for any f∈H1(?), we have
ii. The Second Method
Notice thatwe need only to estimate |a(u ? uh,Πhw) ? ah(u ? uh,Πh?w)|.
Theorem 0.2 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied, and let u∈H01(?)∩H3(?) be the solution to (0-1), anduh∈Uh to (0-6), then we have
II. The Superconvergence For The Isoparametric Bilinear Finite VolumeElement Methods
As we know, in the error analysis, the determination of the upper bound of ||u ?uh||m is usually reduced to the estimation of ||u ?Πhu||m and a(u ?Πhu,Πh?wh). Bythe approximation theory, in general we can only obtain, limited by degree k of theapproximate polynomials, that
In general this estimate can not be improved even if the solution u possesses an highersmoothness. Therefore,
is the optimal order error estimate. But this face does not exclude the possibility thatthe approximation of the derivatives may be of higher order accuracy at some specialpoints, called optimal stress points.
Just as in the case of finite element methods, we can also obtain superconvergenceresults for finite volume methods, provided we manage to get the super interpolationweak estimates. In detail we have the following theorem.
Lemma 0.5 Let u and uh be solutions of the boundary value problem andits finite volume element scheme, respectively. Assume the bilinear form of the finitevolume element scheme satisfies the following interpolation weak estimate: There existsp∈[1,∞] such that then one has
Moreover, let Nk be the set of interpolation optimal stress points: For any P∈Nk,there exists q∈[1,+∞] such thatThen one has ?where r is the number of points in Nk.
To get the superconvergence, the partition Th has to satisfy the h2 uniform con-dition, which also be used in the finite element methods.
Call that the quadrilateral partition Th is h2 uniform, if any two h2 quasi-parallelquadrilateral elements K1 = P1P2P3P4, K2 = P4P3P5P6, which share the commonedge P3P4, satisfy
Under the h2 uniform condition, we prove the first weak estimate of interpola-tion to 2 order, further more, we get the superconvergence of the solutions of thefinite volume element methods at three type points, at which the interpolation possessuperconvergence under the average gradient norm.
Theorem 0.3 Suppose that Th is a h2 uniform quadrilateral partition, let u∈H01(?)∩H3(?)∩W2,∞(?), then
Theorem 0.4 Suppose that Th is a h2 uniform quadrilateral partition, let u∈H01(?)∩H3(?)∩W2,∞(?) be the solution to (0-1), uh∈Uh to (0-6), then
consequently
where Si, i = 1,2,3 denote three types of set of optimal of stresses for Uh-interpolation,i.e. the centers of the elements, the inner vertices and the midpoints of inner edges,and denotes the average value of gradient , ri is the number of points in Si.
III. New Formula And Convergence Analysis Of Isoparametric Bi-quadratic Finite Volume Element Methods
For the Lagrange type finite element space, besides the isoparametric bilinearelements, the isoparametric biquadratic elements are also used commonly. The quarterpoints and three-quarter points of edges of one quadrilateral are taken as the nodes ofthe dual partition in existing formula. It seems reasonable to choose this, because heformula guarantee 2 order convergence in H1 norm. However, they are not the optimalscheme, since the convergence order is only 2 in L2 norm. We choose a new plan toget dual partition, construct a new FVEM formula, and prove the optimal H1 and L2error estimates. Moreover, we present detailed numerical experiments.
For any K∈Th, with vertices Pi,i = 1,2,3,4, taking points Pij, such that
Then the control volumes in K of a vertex Pi, a midpoint Mi and averaging center Qare the subregions PiPi(i+1)PiPi(i?1), Pi(i+1)P(i+1)iPi+1Pi and P1P2P3P4 respectively.
Then for each vertex P, we associate a control volume KP?, which is built by theunion of the above subregions, sharing the vertex P. For each midpoint M on anelement edge, we associate a control volume KM?, which is built by the union of theabove subregions, sharing a common edge including M. The set of all the controlvolumes makes the dual partition Th ? .
The trial function space Uh is taken as isoparametric biquadratic finite elementspace on quadrilateral partition, and the test function space Vh is defined as piecewiseconstant space on dual partition. The finite volume method for (0-1) is : find uh∈Uhsuch that where
Firstly, we obtain the coercivity of the bilinear form.
Lemma 0.6 Suppose that every element in Th satisfies the regular conditionand h2 quasi-parallel quadrilateral condition. Then for su?ciently small h, there exista constant C0 > 0 such that
Further more, we prove the H1 error estimate.
Theorem 0.5 Suppose that u∈H01(?)∩H3(?) is the solution to equation (0-1), uh is the solution to the finite volume formula (y:2:1.2.9), If conditions (0-4), (0-5)and (0-7) hold, then there exist a positive constant C independent of h such that
We also get the optimal L2 error estimates.
Theorem 0.6 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied. Let u∈H01(?)∩H4(?) and uh∈Uh are the solutions to (0-1)and(0-6) respectively, then
IV. Superconvergence Analysis Of Isoparametric Biquadratic Finite Vol-ume Element Methods on rectangular meshes
As we know, the four Gauss points in every quadrilateral are the optimal pointsof stresses for isoparametric biquadratic Uh-interpolation. the solutions of the finiteelement methods are superconvergent at these optimal points of stresses. In this paper,we prove that the derivative of the solutions for the isoparametric biquadratic FVEMare also superconvergent at these points. Numerical experiments show that the super-convergence also holds on quadrilateral meshes. But we can’t prove it with the methodused on rectangular meshes.
Firstly, we introduce Legendre orthogonal polynomials on E = (?1,1)
When i = j, we have (li,lj) = 0, and (lj,lj) = 2j2+1, lj(±1) = (±1)j. We obtain theM type polynomials from the integral on lj
Obviously, if n 2, Mn(±1) = 0. And if j ? i = 0 or±2, (Mi,Mj) = 0, (Mi,Mj) = 0otherwise.
To study the superconvergence of the finite volume methods for 2nd order ellipticequations, we will construct the M type expansion polynomials on square K = [?1,1]×[?1,1]. Suppose that u is smooth enough. Fix t, expand u on s to obtainthen for every bi(t), we take M type expand on twith coe?cients
So, we can obtain the M type expansion for u with these coe?cients
Lemma 0.7 Suppose u∈W4,p(?), 1 p∞, then we have the following estimates of the interpolation uIwhere, Z is the set of the Gauss points in element.
Theorem 0.7 Suppose that Th is a regular rectangular partition, let u∈H01(?)∩H4(?), and uI be the orthogonal interpolation polynomial of u, then
We have the following superclose estimate:
Theorem 0.8 Let Th be a regular rectangular partition, If u∈H01(?)∩H4(?)and uh∈Uh are the solutions to (0-1) and (0-18) respectively, thenfurther
Where S is the set of the optimal points of stresses for Uh interpolation, i.e. the Gausspoints in element. denotes the average value of gradient , r is the number of pointsin S.
引文
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[3] Bi C J. Superconvergence of mixed covolume method for elliptic problems ontriangular grids[J]. J. Comput. Appl. Math., 2008,216(2):534-544.
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[5] Cai Z, Douglas J, Park M. Development and analysis of higher order finite volumemethods over rectangles for elliptic equations[J]. Adv. Comp. Math., 2003,19:3-33.
[6] Cai Z, McCormick S. On the accuracy of the finite volume element method fordi?usion equations on composite grids[J]. SIAM J. Numer. Anal. 1990,27:636-655.
[7] Cai Z, Mandel J, Mccormick S. The finite volume element for di?usion equationson general triangulations[J]. SIAM J. Numer. Anal., 1991,28: 392-402.
[8] Carstensen C, Lazarov R, Tomov S. Explcit and averaging a posteriori error esti-mates for adaptive finite volume methods[J]. SIAM J. Numer. Anal., 2005,42:2496-2521.
[9] Chatzipantelidis P. A finite volume method based on the Crouzeix-Raviart elementfor elliptic PDE’s in two dimensions[J]. Numer. Math., 1999,82: 409-432.
[10] Chatzipantelidis P. Finite Volume Methods for Elliptic PDE’s: A new Approach[J].M2AN, 2002,36: 307-324.
[11]陈传淼.有限元超收敛构造理论[M].长沙:湖南科学技术出版社, 2001.
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[13] Chen Z Y. L2 estimates of linear element generalized di?erence schemes[J]. ActaSci. Natur. Univ. Sunyaseni, 1994(4): 22-28.
[14] Chen Z Y. Superconvergence of generalized di?erence methods for elliptic bound-ary value problem[J]. Numer. Math. -J. Chinese Univ., English Series, 1994(3):163-171.
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[17] Chen Z Y. The error estimate of generalized di?erence method 3rd-order Hermitetype for elliptic partial di?erential equations[J]. Northeast. Math. J., 1992(8): 127-135.
[18] Chen Z Y, Li R H, Zhou A H. A note on the optimal estimate of the finite volumeelement method[J]. Adv. Comp. Math., 2002(16): 291-302.
[19] Chou S H, Kwak D Y. A covolume method based on rotated bilinears for thegeneralized Stokes problem[J]. SIAM J. Numer. Anal., 1998,35: 494-507.
[20] Chou S H, Kwak D Y. Multigrid algorithms for a vertex-centered covolume methodfor elliptic problems[J]. Numer. Math., 2002,90: 441-458.
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[29] Chou S H, Ye X. Unified analysis of finite volume methods for second order ellipticproblems[J]. SIAM J. Numer. Anal., 2007(45): 1639-1653.
[30] Ciarlet P G. The finite element methods for elliptic problems[M]. NorthHolland:Amesterdam, 1978.
[31] Croisille J P. Finite volume box-schemes and mixed methods[J]. Math. Model.Numer. Anal., 2000,34(5):1087-1106.
[32] Droniou J. Error estimates for the convergence of a finite volume discretization ofconvection-di?usion equations[J]. J. Numer. Math., 2003,11:1-32.
[33] Ewing R E, Lin T, Lin Y. On the accuracy of finite volume element method basedon piecewise linear polynomials[J]. SIAM J. Numer. Anal., 2002(39): 1865-1888.
[34] Ewing R E, Liu M M, Wang J P. Superconvergence of mixed finite element ap-proximations over quadrilaterals[J]. SIAM J. Numer. Anal., 1999,36(3):772-787.
[35] Eymard R, Gallou¨et T, Herbin R. A finite volume for anisotropic di?usion prob-lems[J]. Comptes Rendus l’acad. Sci., 2004,339:299-302.
[36] Faille I. Acontrol volume method to solve an elliptic equation on a two-dimensionalirregular mesh[J]. Comput. Methods Appl. Mech. Eng., 1992,100:275-290.
[37]丰连海.求解二阶椭圆型偏微分方程的一种有限体积元格式[J]. 2002(4):63-67.
[38] Girault V. Theory of a finite di?erence method on irregular networks[J]. SIAMJ.Numer.Anal., 1974(11): 260-282.
[39] Girault V. Raviart P A. The finite element methods for Navier-Stokes equations:theory and algorithms[M]. Springer-Verlag: Berlin, 1986.
[40] Hackbusch W. On first and second order box schemes[J]. Computing, 1989,41:277-296.
[41] Hu J, Shi Z C. Constrained quadrilateral nonconforming rotated Q1 element[J].J. Comp. Math., 2005,23(6): 561-586.
[42] Huang H T, Li Z C, Zhou A. New error estimates of biquadratic Lagrange elementsfor Poisson’s equation[J]. Applied Numerical Mathematics, 2006,56:712-744.
[43] Huang J G, Xi S T. On the finite volume element method for general self-adjointelliptic problem, SIAM J. Numer. Anal., 1998(35): 1762-1774.
[44] Huang J G, Li L K. Some superconvergence results for the covolume method forelliptic problems[J]. Commun. Numer. Meth. Engng, 2001,17:291-302.
[45]金继承,黄云清.双二次有限元解导数的超收敛性[J].湘潭大学自然科学学报, 1993,15(2):8-14.
[46] Kim K Y. Error estimates for a mixed finite volume method for the p-laplacianproblem[J]. Numer. Math., 2005,101:121-142.
[47] Kim K Y. New mixed finite volume methods for second order elliptic problems[J].ESAIM: M2AN., 2006,40(1):123-147.
[48] K?r′??zek M., Neittaanm¨aki P. On superconvegence techniques[J]. Acta Appl. Math.1987,9:175-198.
[49] Lazarov R, Michev I, Vassilevski P. Finite volume methods for convec-tion–di?usion problems[J]. SIAM J. Numer. Anal. 1996,33:31-55.
[50] Lesaint P, Zlamal M. Superconvergence of the gradient of finite element solu-tions[J]. R.A.I.R.O. Numer. Anal., 1979(12): 139-166.
[51] Li Q, Jiang Z. Optimal maximum norm estimates and superconvergence for GDMon two-point boundary value problems[J]. Northeast Math J., 1999(1): 89-96.
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[53]李荣华,祝丕琦.二阶椭圆偏微分方程的广义差分法(I)―三角网情形[J].高校计算数学学报, 1982(2): 140-152 .
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[60] Liebau, F. The Finite volume element method with quadratic basis functions[J].Computing, 1996,57: 281-299.
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