几种求解H（curl）与H（grad）型偏微分方程有限元离散系统的多水平快速算法
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摘要
H(curl)型和H(grad)型偏微分方程组(PDEs)在电磁场计算和固体力学等领域具有广泛应用.本文针对频域电磁场PML逼近方程组、频域电磁场隐身超常材料模型和不定时谐Maxwell方程组等几类重要的H(curl)型偏微分方程组的棱有限元离散系统,以及Possion方程和线弹性力学方程组等几类重要的H(grad)型偏微分方程的间断Galerkin (DG)有限元离散系统和二次协调有限元离散系统,研究了相应的多水平快速算法.主要内容如下:
针对一类二维电磁场散射问题PML逼近方程的对称不定线性棱有限元离散系统,首次利用迭代两网格法的思想为其构造了一种迭代两网格法,该方法本质地将对原复杂离散系统的求解转化为三个更为简单的辅助系统的求解.同时为其中的两个辅助系统分别构造了高效预条件子,以及基于相应预条件子的PMINRES方法和PCG方法.数值实验表明,与经典的迭代法(如PMINRES法)相比,我们设计的迭代两网格法在迭代次数和CPU时间上均具有明显的优势.
通过将完美匹配层(PML)方法、电磁场理论以及隐身超常材料理论等相结合,给出了一种二维频域电磁场隐身超常材料模型,数值实验检验了模型的正确性.进一步,为一类电磁场散射问题PML逼近方程组和上述隐身超常材料模型方程组的线性棱有限元离散系统,结合HX预条件方法,构造了一种Schur补型的多水平预条件子,数值实验验证了算法的有效性.
针对不定时谐Maxwell方程组的棱有限元离散系统,构造了一种基于两水平方法和HX预条件子的多水平加性预条件子,本质性地将细网格空间上原不定问题的求解转化为粗网格空间和细网格上curl算子核空间上原不定问题的求解,以及在细网格空间上的对称正定问题的求解.当粗网格尺寸充分小时,证明了基于该预条件子的PGMRES法的一致收敛性.数值实验验证了理论结果的正确性.
针对一类二阶椭圆问题的一般对称内罚间断有限元离散系统,通过将DG有限元空间分解为“高频”部分和线性协调元空间,为其构造一种新的稳定的空间分解.利用该空间分解,首先为DG有限元离散系统设计了多水平预条件子.基于辅助空间预条件理论,我们证明了该预条件系统的条件数是一致有界的,同时为其中的线性元系统设计了BPX预条件子,从而得到一种多水平加性预条件子.接着还为DG有限元离散系统设计了一种两水平迭代法,并证明该迭代法是一致收敛的.数值实验验证了上述理论的正确性.
针对平面弹性力学问题二次有限元离散系统,给出了一种基于二分网格的局部多层网格(LMG)法.通过将二次有限元空间分解成‘高频’部分和线性元空间,结合二分网格和插值算子的特性,证明了该空间分解的稳定性和强Cauchy-Schwarz不等式成立,进而得到了LMG算法的一致收敛结果.数值实验验证了理论的正确性.
H(curl) and H(grad) partial differential equations (PDEs) have been widely appliedin the fields of calculation electromagnetic and solid mechanics. In this paper, we discussthe multilevel fast algorithms for the edge finite element discretizations of H(curl) PDEs,such as the frequency domain PML approximations equations, frequency domain modelabout electromagnetic cloak metamaterials and the indefinite time-harmonic Maxwell’sequations, and for the discontinuous Galerkin (DG) discretizations and quadric finite ele-ment discretizations related to Possion and linear elasticity equations. The main contentsare as follows.
Using the idea of iterative two-grid method, we propose firstly an iterative two-gridmethod for the symmetry and indefinite edge finite element discretizations of PML equa-tionstotheMaxwellscatteringproblemintwodimensions. Bythismethod,weessentiallytransform the complex original problem in a fine space into three more simple auxiliarysystems. Further, we construct efficient preconditioners for two of these systems andfast iterative methods, such as PMINRES and PCG, based on the corresponding precon-ditioners. Compared with some iterative methods to solve saddle-point systems, suchas PMINRES, numerical experiments show the competitive performance of our iterativetwo-grid method.
By using perfect matched layer(PML) technical and the theories of electromagneticfields and cloaking metamaterials, we develop a frequency domain model about electro-magnetic cloak metamaterials in two spatial dimension. Numerical experiments verifythe correctness of the model. Furthermore, based on a Schur-type preconditioner and HXpreconditioner, we construct multilevel predconditioner for the edge finite element dis-cretizations about a PML approximation equations for Maxwell scattering problems andthe above cloaking metamaterials model, respectively. Numerical experiments show thatour multilevel algorithms are efficiency.
A multilevel additive preconditioner based on two level method and HX prdcon-ditioner for edge discretizations of the indefinite time-harmonic Maxwell’s equations isconstructed, which essentially translates the computation of the original problem in thefine mesh space into the the computation of the original problem in the kernel of thecurl-operator in the fine mesh space and a coarse mesh space, and the computation of acorresponding symmetric positive definite problem in the fine mesh space. We also prove the uniform convergence of PGMRES method based on the multilevel additive precondi-tioner, if the coarse mesh size is sufficiently small. Numerical experiments indicate thetheory.
By decomposing the DG finite element space into a space containing the high fre-quency components and a linear conforming space, we construct a new stable space de-composition for the discretization of second order elliptic problems by the symmetricdiscontinuous Galerkin methods. By using this new decomposition, we develop a multi-level additive preconditioner for DG discretization, prove that the condition numbers ofthe preconditioned system are uniformly bounded based on the auxiliary space theory,and develop a BPX preconditioner for the linear system on a conforming space. Further-more, we develop an two level iterative method for the DG discretization and prove thatthe iterative method is uniformly convergent. Numerical experiments are also shown toconfirm these theoretical results.
We develop a local multigrid method (LMG) based on bisection grids for quadricfinite element discretizations of plane elasticity problems in two spatial dimension. Bydecomposing the quadric finite space into‘high frequency’and linear finite space, andusing the properties of bisection grids and interpolation operators, we show this decom-position is stable and satisfies strong Cauchy-Schwarz inequality. Furthermore we provethat the LMG method converges uniformly with respect to the mesh size. Numerical ex-periments confirm the theory.
针对一类二维电磁场散射问题PML逼近方程的对称不定线性棱有限元离散系统,首次利用迭代两网格法的思想为其构造了一种迭代两网格法,该方法本质地将对原复杂离散系统的求解转化为三个更为简单的辅助系统的求解.同时为其中的两个辅助系统分别构造了高效预条件子,以及基于相应预条件子的PMINRES方法和PCG方法.数值实验表明,与经典的迭代法(如PMINRES法)相比,我们设计的迭代两网格法在迭代次数和CPU时间上均具有明显的优势.
通过将完美匹配层(PML)方法、电磁场理论以及隐身超常材料理论等相结合,给出了一种二维频域电磁场隐身超常材料模型,数值实验检验了模型的正确性.进一步,为一类电磁场散射问题PML逼近方程组和上述隐身超常材料模型方程组的线性棱有限元离散系统,结合HX预条件方法,构造了一种Schur补型的多水平预条件子,数值实验验证了算法的有效性.
针对不定时谐Maxwell方程组的棱有限元离散系统,构造了一种基于两水平方法和HX预条件子的多水平加性预条件子,本质性地将细网格空间上原不定问题的求解转化为粗网格空间和细网格上curl算子核空间上原不定问题的求解,以及在细网格空间上的对称正定问题的求解.当粗网格尺寸充分小时,证明了基于该预条件子的PGMRES法的一致收敛性.数值实验验证了理论结果的正确性.
针对一类二阶椭圆问题的一般对称内罚间断有限元离散系统,通过将DG有限元空间分解为“高频”部分和线性协调元空间,为其构造一种新的稳定的空间分解.利用该空间分解,首先为DG有限元离散系统设计了多水平预条件子.基于辅助空间预条件理论,我们证明了该预条件系统的条件数是一致有界的,同时为其中的线性元系统设计了BPX预条件子,从而得到一种多水平加性预条件子.接着还为DG有限元离散系统设计了一种两水平迭代法,并证明该迭代法是一致收敛的.数值实验验证了上述理论的正确性.
针对平面弹性力学问题二次有限元离散系统,给出了一种基于二分网格的局部多层网格(LMG)法.通过将二次有限元空间分解成‘高频’部分和线性元空间,结合二分网格和插值算子的特性,证明了该空间分解的稳定性和强Cauchy-Schwarz不等式成立,进而得到了LMG算法的一致收敛结果.数值实验验证了理论的正确性.
H(curl) and H(grad) partial differential equations (PDEs) have been widely appliedin the fields of calculation electromagnetic and solid mechanics. In this paper, we discussthe multilevel fast algorithms for the edge finite element discretizations of H(curl) PDEs,such as the frequency domain PML approximations equations, frequency domain modelabout electromagnetic cloak metamaterials and the indefinite time-harmonic Maxwell’sequations, and for the discontinuous Galerkin (DG) discretizations and quadric finite ele-ment discretizations related to Possion and linear elasticity equations. The main contentsare as follows.
Using the idea of iterative two-grid method, we propose firstly an iterative two-gridmethod for the symmetry and indefinite edge finite element discretizations of PML equa-tionstotheMaxwellscatteringproblemintwodimensions. Bythismethod,weessentiallytransform the complex original problem in a fine space into three more simple auxiliarysystems. Further, we construct efficient preconditioners for two of these systems andfast iterative methods, such as PMINRES and PCG, based on the corresponding precon-ditioners. Compared with some iterative methods to solve saddle-point systems, suchas PMINRES, numerical experiments show the competitive performance of our iterativetwo-grid method.
By using perfect matched layer(PML) technical and the theories of electromagneticfields and cloaking metamaterials, we develop a frequency domain model about electro-magnetic cloak metamaterials in two spatial dimension. Numerical experiments verifythe correctness of the model. Furthermore, based on a Schur-type preconditioner and HXpreconditioner, we construct multilevel predconditioner for the edge finite element dis-cretizations about a PML approximation equations for Maxwell scattering problems andthe above cloaking metamaterials model, respectively. Numerical experiments show thatour multilevel algorithms are efficiency.
A multilevel additive preconditioner based on two level method and HX prdcon-ditioner for edge discretizations of the indefinite time-harmonic Maxwell’s equations isconstructed, which essentially translates the computation of the original problem in thefine mesh space into the the computation of the original problem in the kernel of thecurl-operator in the fine mesh space and a coarse mesh space, and the computation of acorresponding symmetric positive definite problem in the fine mesh space. We also prove the uniform convergence of PGMRES method based on the multilevel additive precondi-tioner, if the coarse mesh size is sufficiently small. Numerical experiments indicate thetheory.
By decomposing the DG finite element space into a space containing the high fre-quency components and a linear conforming space, we construct a new stable space de-composition for the discretization of second order elliptic problems by the symmetricdiscontinuous Galerkin methods. By using this new decomposition, we develop a multi-level additive preconditioner for DG discretization, prove that the condition numbers ofthe preconditioned system are uniformly bounded based on the auxiliary space theory,and develop a BPX preconditioner for the linear system on a conforming space. Further-more, we develop an two level iterative method for the DG discretization and prove thatthe iterative method is uniformly convergent. Numerical experiments are also shown toconfirm these theoretical results.
We develop a local multigrid method (LMG) based on bisection grids for quadricfinite element discretizations of plane elasticity problems in two spatial dimension. Bydecomposing the quadric finite space into‘high frequency’and linear finite space, andusing the properties of bisection grids and interpolation operators, we show this decom-position is stable and satisfies strong Cauchy-Schwarz inequality. Furthermore we provethat the LMG method converges uniformly with respect to the mesh size. Numerical ex-periments confirm the theory.
引文
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