高波数Helmholtz方程的高阶连续多罚有限元方法的稳定性估计（英文）
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摘要
本文考虑二维和三维区域上高波数Helmholtz散射问题的高阶(多项式次数p≥2)连续多罚有限元方法.本文证明在加罚参数的虚部大于零的条件下,对任意k, h, p,连续多罚有限元方法是绝对稳定的,即都存在唯一解.这里k是波数, h为网格尺寸.
Some continuous interior multi-penalty finite element method(CMP-FEM) of using piecewise polynomials of order p 2 for the Helmholtz equation in the two and three dimensions is considered. The proposed CMP-FEM is stable(hence well-posed) for any k, h, p and penalty parameters with positive imaginary parts, where k is the wave number,h is the mesh size.
Some continuous interior multi-penalty finite element method(CMP-FEM) of using piecewise polynomials of order p 2 for the Helmholtz equation in the two and three dimensions is considered. The proposed CMP-FEM is stable(hence well-posed) for any k, h, p and penalty parameters with positive imaginary parts, where k is the wave number,h is the mesh size.
引文
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[30]FENG Xiaobing,KARAKASHIAN O.A.Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition[J].Math.Comp.,2007,76:1093-1117.
[31]FENG Xiaobing,XING Y.Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number[J].Mathematics of Computation,2013,82(4):2872-2896.
[32]SHEN Jie,WANG L.Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains[J].SIAM J.Numer.Anal.,2007,45(5):1954-1978.
[33]SCHWAB C.p-and hp-Finite Element Methods[M].UK,Oxford:Oxford University Press,1998.
[2]DOUGLAS J J,SANTOS J E,SHEEN D.Approximation of scalar waves in the space-frequency domain[J].Math.Models Methods Appl.Sci.,1994,4:509-531.
[3]DOUGLAS J J,DUPONT T.Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods[M].Berlin:Springer-Verlag,Lecture Notes in Phys.58,1976.
[4]BURMAN Erik.A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty[J].SIAM J.Numer.Anal.,2005,43(5):2012-2033.
[5]BURMAN Erik,ERM A.Stabilized Galerkin approximation of convection-diffusion-reaction equations:discrete maximum principle and convergence[J].Math.Comp.,2005,74(252):1637-1652.
[6]BURMAN Erik,ERN A.Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations[J].Math.Comp.,2007,259:1119-1140.
[7]BURMAN Erik,FERN′ANDEZ M,HANSBO P.Continuous interior penalty finite element method for Oseen’s equations[J].SIAM J.Numer.Anal.,2006,44(3):1248-1274.
[8]BURMAN Erik,HANSBO P.Edge stabilization for Galerkin approximations of convection-diffusionreaction problems[J].Comput.Methods Appl.Mech.Engrg.,2004,193(15-16):1437-1453.
[9]WU Haijun.Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number.Part I:Linear version[J].IMA J.Numer.Anal.,2014,34:1266-1288.
[10]ZHU Lingxue,WU Haijun.Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number.Part II:hp Version[J].SIAM J.Numer.Anal.,2013,51:1828-1852.
[11]DU Yu,WU Haijun.Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number[J].SIAM Journal on Numerical Analysis,2015,53:782-804.
[12]BURMAN Erik,WU Haijun,ZHU Lingxue.Linear continuous interior penalty finite element method for Helmholtz equation with high wave number:One dimensional analysis[J].Numerical Methods for Partial Differential Equations,2016,32:1378-1410.
[13]ARNOLD D.An interior penalty finite element method with discontinuous elements[J].SIAM J.Numer.Anal.,1982,19:742-760.
[14]BRENNER S,SUNG L.C0interior penalty methods for fourth order elliptic boundary value problems on polygonal domains[J].Journal of Scientific Computing,2005,22(1):83-118.
[15]FENG Xiaobing,WU Haijun.Discontinuous Galerkin methods for the Helmholtz equation with large wave numbers[J].SIAM J.Numer.Anal.,2009,47(4):2872-2896.
[16]FENG Xiaobing,WU Haijun.hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number[J].Math.Comp.,2011,80(276):1997-2024.
[17]ZHU Lingxue,DU Yu.Pre-asymptotic error analysis of hp-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number[J].Computers and Mathematics with Applications,2015,70:917-933.
[18]DU Yu,ZHU Lingxue.Preasymptotic error analysis of high order interior penalty discontinuous galerkin methods for the helmholtz equation with highwave number[J].Journal of Scientific Computing,2016,67:130-152.
[19]MELENK J.On Generalized Finite Element Methods[D].Maryland:University of Maryland at College Park,1995.
[20]CUMMINGS P,FENG Xiaobing.Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations[J].Math.Models Methods Appl.,2006,16(1):139-160.
[21]HETMANIUK U.Stability estimates for a class of Helmholtz problems[J].Commun.Math.Sci.,2007,5(3):665-678.
[22]KARAKASHIAN O A,Pascal F.A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems[J].SIAM J.Numer.Anal.,2003,41(6),2374-2399.
[23]OSWALD P.On a BPX-preconditioner for p1 elements[J].Computing,1993,51(2):125-133.
[24]BRENNER S,SCOTT L.The Mathematical Theory of Finite Element Methods[M].3rd ed,Berlin:Springer-Verlag,2008.
[25]CIARLET P G.The Finite Element Method for Elliptic Problems[M].Amsterdam:North-Holland,1978.
[26]MELENK J M,SAUTER S.Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[J].Math.Comp.,2010,79(272):1871-1914.
[27]MELENK J M,SAUTER S.Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation[J].SIAM J.Numer.Anal.,2011,49(3):1210-1243.
[28]BABUˇSKA I,ZL′AMAL M.Nonconforming elements in the finite element method with penalty[J].SIAM J.Numer.Anal.,1973,10(5):863-875.
[29]BAKER G.Finite element methods for elliptic equations using nonconforming elements[J].Math.Comp.,1977,31:44-59.
[30]FENG Xiaobing,KARAKASHIAN O.A.Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition[J].Math.Comp.,2007,76:1093-1117.
[31]FENG Xiaobing,XING Y.Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number[J].Mathematics of Computation,2013,82(4):2872-2896.
[32]SHEN Jie,WANG L.Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains[J].SIAM J.Numer.Anal.,2007,45(5):1954-1978.
[33]SCHWAB C.p-and hp-Finite Element Methods[M].UK,Oxford:Oxford University Press,1998.