不规则区域上椭圆问题的配置有限元法
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摘要
配置法是近二、三十年发展起来的以满足纯插值约束条件的方式,寻求算子方程近似解的数值方法,具有无需计算数值积分,计算简便及收敛精度高等优点,使之在工程技术和计算数学的许多领域得到广泛的应用。配置法是通过分片多项式求近似解,使之在某些特定的点即配置点上满足微分方程及其边界条件。最初样条配置法是利用三次样条函数并在自然节点上进行配置,但精度不够高。为了加速收敛速度,采用高斯数值积分公式的节点代替自然节点进行配置,且选用分片双三次Hermite插值多项式空间作为求解的函数逼近空间,收敛速度可达到h~4阶,并称在高斯节点上的样条配置法为正交样条配置法(OSC方法)。
正交样条配置法最初是由C.deBoor和Swartz[2]提出的,考虑的是m阶常微分方程。在一维情况F,Douglas和Dupont[3]对抛物方程提出C~1有限元配置方法(R≥3)。在二维情况下,Prenter和Rusell[6]考虑了椭圆方程的OSC方法,Bialecki和Cai[11]对椭圆方程的边界考虑了两种插值技巧,即Hermite插值和Gauss插值,都得到了最优估计。Percell和Wheeler[5]研究了R≥3情况下的椭圆问题。Bialecki[12]扩展并概括了二维椭圆边值问题的理论结果,且在[13]中得到超收敛结果.这里的R表示配置多项式的次数。
正交配置法较之有限元法易于实现精度高,原因在于配置法不需要计算数值积分,而数值积分既要增加工作量,又要影响系数矩阵的精度,所以配置法被广泛应用于数学物理及工程问题。
Emil(?).Frind和George F.Pinder[1]研究了不规则区域Laplace方程的配置有限元方法,这种方法把正交配置法和有限元方法结合起来,运用了在每个节点具有四个自由度的双三次Hermite元,同时运用亚参数变换表示不规则区域,经过亚参数变换,配置点的位置能够确定,从而得到最大的精确度。这种方法特别适合位势问题,这里要求解是C~1连续的。本文的主要工作是把这种数值方法应用到更为一般的椭圆方程。
全文共分为三章。
第一章主要以引理的形式给出Laplace方程不规则区域配置法的基础理论;第二章讨论了不规则区域一类椭圆边值问题的配置有限元方法,提出问题并根据引理求解;第三章给出算例,并对配置法和Galerkin有限元法的计算效率进行对比,最后给出结论以及本文的不足。
The collocation method is a munerical method which searches for the approximation solution of the operator function by satisfying pure interpolation condition for about thirty years,and it is widely used for solving both engineering and computing mathematics due to its ease of implementation,high-order accuracy and no integrals need be evaluated.Collocation method essentially involves determining an approximate solution by a piecewise polynomial by requiring it to satisfy the differential equation and boundary conditions exactly at certain points.Original spline collocation methods collocate at the nodes by cubic spline functions,but the precision isn't good.For improving the convergence rate,collocation points usually use the nodes of Gauss quadrature formula,and choose piecewise Hermite bicubic polynomials as the approximative space,convergence rate can reach h~4,spline collocation at Gauss points is named orthogonal spline collocation method(OSC).
Orthogonal collocation method was first introduced by C.deBoor and Swartz[2] for m-th order ODE.In one space variable,Douglas and Dupont[3]give C~1 finite element method(R≥3).In two space variables,Prenter and Rusell[6]gives OSC for elliptic equation.Bialecki and Cai[11]consider two kinds of interpolation for the boundary conditions of elliptic equation,i.e.Hermite intetpolation and Gauss interpolation, optimal estimate can be get.Percell and Wheeler[5]consider the elliptic problems(R≥3).Bialeki[12]extends and generalizes the theoretical result for elliptic problems,and in[13]gets superconvergence result.
Orthogonal collocation method is higher convergence rate than finite element ,since OSC needn't compute integrals,numerical integrals increase the workload and effect the precision of coefficient matrix.Then collocation methods are used widely for mathematical physics snd engineering problems.
Emil O.Frind and George F.Pinder(?)1 gave a collocation finite element method for Laplace equation in irregular domains.The main work of this thesis is to extend this method to elliptic problems.In this thesis,a potentially powerful numerical method for solving elliptic boundary value problems in irregular domains is proposed.The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements.Bicubic Hermite elements with four degreesof-freedom per node are used.A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representatin of irregular boundaries.
This thesis is divided into three chapters.
In chapter one,we give base theory of collocation element,method for Laplace equation in irregular domains and give three lemmas.In chapter two,we give collocation finite element method for elliptic equation in irregular domains:In chapter three,we give two numerical examples,and give a comparison between orthogonal collocation and Galerkin finite elements in computational efficiency.In the end,we give the conclusion and the shortage of this thesis.
正交样条配置法最初是由C.deBoor和Swartz[2]提出的,考虑的是m阶常微分方程。在一维情况F,Douglas和Dupont[3]对抛物方程提出C~1有限元配置方法(R≥3)。在二维情况下,Prenter和Rusell[6]考虑了椭圆方程的OSC方法,Bialecki和Cai[11]对椭圆方程的边界考虑了两种插值技巧,即Hermite插值和Gauss插值,都得到了最优估计。Percell和Wheeler[5]研究了R≥3情况下的椭圆问题。Bialecki[12]扩展并概括了二维椭圆边值问题的理论结果,且在[13]中得到超收敛结果.这里的R表示配置多项式的次数。
正交配置法较之有限元法易于实现精度高,原因在于配置法不需要计算数值积分,而数值积分既要增加工作量,又要影响系数矩阵的精度,所以配置法被广泛应用于数学物理及工程问题。
Emil(?).Frind和George F.Pinder[1]研究了不规则区域Laplace方程的配置有限元方法,这种方法把正交配置法和有限元方法结合起来,运用了在每个节点具有四个自由度的双三次Hermite元,同时运用亚参数变换表示不规则区域,经过亚参数变换,配置点的位置能够确定,从而得到最大的精确度。这种方法特别适合位势问题,这里要求解是C~1连续的。本文的主要工作是把这种数值方法应用到更为一般的椭圆方程。
全文共分为三章。
第一章主要以引理的形式给出Laplace方程不规则区域配置法的基础理论;第二章讨论了不规则区域一类椭圆边值问题的配置有限元方法,提出问题并根据引理求解;第三章给出算例,并对配置法和Galerkin有限元法的计算效率进行对比,最后给出结论以及本文的不足。
The collocation method is a munerical method which searches for the approximation solution of the operator function by satisfying pure interpolation condition for about thirty years,and it is widely used for solving both engineering and computing mathematics due to its ease of implementation,high-order accuracy and no integrals need be evaluated.Collocation method essentially involves determining an approximate solution by a piecewise polynomial by requiring it to satisfy the differential equation and boundary conditions exactly at certain points.Original spline collocation methods collocate at the nodes by cubic spline functions,but the precision isn't good.For improving the convergence rate,collocation points usually use the nodes of Gauss quadrature formula,and choose piecewise Hermite bicubic polynomials as the approximative space,convergence rate can reach h~4,spline collocation at Gauss points is named orthogonal spline collocation method(OSC).
Orthogonal collocation method was first introduced by C.deBoor and Swartz[2] for m-th order ODE.In one space variable,Douglas and Dupont[3]give C~1 finite element method(R≥3).In two space variables,Prenter and Rusell[6]gives OSC for elliptic equation.Bialecki and Cai[11]consider two kinds of interpolation for the boundary conditions of elliptic equation,i.e.Hermite intetpolation and Gauss interpolation, optimal estimate can be get.Percell and Wheeler[5]consider the elliptic problems(R≥3).Bialeki[12]extends and generalizes the theoretical result for elliptic problems,and in[13]gets superconvergence result.
Orthogonal collocation method is higher convergence rate than finite element ,since OSC needn't compute integrals,numerical integrals increase the workload and effect the precision of coefficient matrix.Then collocation methods are used widely for mathematical physics snd engineering problems.
Emil O.Frind and George F.Pinder(?)1 gave a collocation finite element method for Laplace equation in irregular domains.The main work of this thesis is to extend this method to elliptic problems.In this thesis,a potentially powerful numerical method for solving elliptic boundary value problems in irregular domains is proposed.The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements.Bicubic Hermite elements with four degreesof-freedom per node are used.A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representatin of irregular boundaries.
This thesis is divided into three chapters.
In chapter one,we give base theory of collocation element,method for Laplace equation in irregular domains and give three lemmas.In chapter two,we give collocation finite element method for elliptic equation in irregular domains:In chapter three,we give two numerical examples,and give a comparison between orthogonal collocation and Galerkin finite elements in computational efficiency.In the end,we give the conclusion and the shortage of this thesis.
引文
[1]Emil O.Frind and George F.Pinder,A Collocation Finite Element Method for Polenlial Problems in Urregular Domains,International Journal For Numerical Methods in Engineering,VOL.14(1979),681-701.
[2]C.dcBoor.B.Swartz,Collocation at Gauss Points,SIAM J.Numer.Anal.,10(1973),582-606.
[3]Jim Douglas.Jr and T.Dupont,Collocation Methods for Parabolic Equations in a Single Space Variable,Lecture Notes in Mathematics,VOL.385,Springer-Verlag ,Hcidelberg.(1974).
[4]B.A.Finlayson,The Method of Weighted Residuals and Variational Principles,Academic Press,NewYork.(1972).
[5]P.Percell,M.F.Wheeler,A C~1 Finite Element Collation Method for Elliptic Equations,SIAM J.Numer.Anal.,12(1980),605-622.
[6]P.M.Prenter,R.D.Russell,Orthogonal Collocation for Elliptic Partial Differential Equations,SIAM J.Numer.Anal.,13(1976),923-939.
[7]P.M.Prcnter,Splines and Variational Methods,from Pure and Applied Methomatics Interscience Series,Wiley,NewYork,(1975).
[8]J.V.Villadsen and W.E.Stewart,Solution of Boundary Value Problems by Orthogonal Collocation,Chem.Eng.Sci.22(1967),1485-1501.
[9]O.Zienkiewicz,The Finite Element Method in Engineering Science,McGraw-Hill ,London,(1971).
[10][美]P.M.Prenter,样条函数与变分方法,上海科学技术出版社,上海,(1980).
[11]B.Bialecki,Xiao-Chuan Cai,II~1-norm Error Bounds for Piecewise Hermite Bicubic Orthogonal Spline Collocation Schemes for Elliptic Boundary Value Prroblems ,SIAM J.Numer.Anal.,31(4)(1994),1128-1146.
[12]B.Bialecki,Convergence Analysis of Orthogonal Spline collocation for Elliptic Boundary Value Problems,SIAM J.Numer.Anal.,35(1998),617-631.
[13]B.Bialecki,Superconvergence of Orthogonal Spline Collocation in the Solution of Poisson's Equation.Numer.Methods Partial Differeential Equations,15(1999),285-303.
[14]B.Bialecki,Preconditioned Richardson and minimal residual iterative methods for piecewise Hermite bicubic orthogonal spline collocation equations.SIAM J.Sci.Comput.,15(1994),668-680.
[15]B.Bialecki,G.Fairweather,and K.R.Bennett,Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations,SIAM J.Numer.Anal.,29(1992),156-173.
[16]B,Bialecki,andR.I.Fernandes,Laplace-madified and alternating-direction orthogonal spline collocation methods for parabolic problems on rectangles,Math.Comp.,60(1993),545-573.
[2]C.dcBoor.B.Swartz,Collocation at Gauss Points,SIAM J.Numer.Anal.,10(1973),582-606.
[3]Jim Douglas.Jr and T.Dupont,Collocation Methods for Parabolic Equations in a Single Space Variable,Lecture Notes in Mathematics,VOL.385,Springer-Verlag ,Hcidelberg.(1974).
[4]B.A.Finlayson,The Method of Weighted Residuals and Variational Principles,Academic Press,NewYork.(1972).
[5]P.Percell,M.F.Wheeler,A C~1 Finite Element Collation Method for Elliptic Equations,SIAM J.Numer.Anal.,12(1980),605-622.
[6]P.M.Prenter,R.D.Russell,Orthogonal Collocation for Elliptic Partial Differential Equations,SIAM J.Numer.Anal.,13(1976),923-939.
[7]P.M.Prcnter,Splines and Variational Methods,from Pure and Applied Methomatics Interscience Series,Wiley,NewYork,(1975).
[8]J.V.Villadsen and W.E.Stewart,Solution of Boundary Value Problems by Orthogonal Collocation,Chem.Eng.Sci.22(1967),1485-1501.
[9]O.Zienkiewicz,The Finite Element Method in Engineering Science,McGraw-Hill ,London,(1971).
[10][美]P.M.Prenter,样条函数与变分方法,上海科学技术出版社,上海,(1980).
[11]B.Bialecki,Xiao-Chuan Cai,II~1-norm Error Bounds for Piecewise Hermite Bicubic Orthogonal Spline Collocation Schemes for Elliptic Boundary Value Prroblems ,SIAM J.Numer.Anal.,31(4)(1994),1128-1146.
[12]B.Bialecki,Convergence Analysis of Orthogonal Spline collocation for Elliptic Boundary Value Problems,SIAM J.Numer.Anal.,35(1998),617-631.
[13]B.Bialecki,Superconvergence of Orthogonal Spline Collocation in the Solution of Poisson's Equation.Numer.Methods Partial Differeential Equations,15(1999),285-303.
[14]B.Bialecki,Preconditioned Richardson and minimal residual iterative methods for piecewise Hermite bicubic orthogonal spline collocation equations.SIAM J.Sci.Comput.,15(1994),668-680.
[15]B.Bialecki,G.Fairweather,and K.R.Bennett,Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations,SIAM J.Numer.Anal.,29(1992),156-173.
[16]B,Bialecki,andR.I.Fernandes,Laplace-madified and alternating-direction orthogonal spline collocation methods for parabolic problems on rectangles,Math.Comp.,60(1993),545-573.