几种求解辐射扩散问题和线弹性问题的代数多层网格法与区域分解法
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摘要
代数多层网格(AMG)法和区域分解法(DDM)是国际上流行的两类求解大规模偏微分方程(PDEs)离散化系统的快速方法.目前,对于求解复杂PDEs离散化系统的AMG法和DDM,还有许多需要进一步研究的问题.本文针对两类具有广泛应用背景的PDEs离散化系统,研究其高效(并行)AMG法和DDM,主要工作如下：
针对一种关于三温辐射扩散方程离散化系统的代数两层预条件子B01,测试并分析了相应的PGMRES法的算法效率.通过引入能刻画系统耦合强弱和单温子系统对角占优强弱的若干因子,设计了一种基于块对角型和PCTL型的自适应预条件子B2.对LARED-S程序产生的数据进行了测试,表明新算法比基于RSAMG和B01预条件子的PGMRES法更加稳健和高效.进一步针对JASMIN接口,为预条件子B2设计了基于进程分组策略的并行实现算法Bp2,通过对源于实际应用背景的数据进行测试,表明所设计的并行解法器比基于BoomerAMG预条件子的PGMRES解法器具有更好的算法可扩展性,且具有更好的运算效率.此外,还为预条件子Bp2提供了串行接口.
针对二维定常扩散问题在两层SAMR网格下的混合五点格式和SFVE格式,分析了数值解的逼近性与解函数在粗细界面附近的性态以及插值算子精度之间的关系,实验结果表明后者具有更好的普适性；同时为混合五点格式和SFVE格式设计两层网格(TL)法,并证明了后者的一致收敛性.针对二维三温辐射扩散问题,为SAMR网格下的混合五点格式和SFVE格式,设计了相应的自适应PCTL预条件子Btl2其中子系统利用TL法求解.与基于RSAMG预条件子的PGMRES法相比,基于Btl2的PGMRES法更适合于在JASMIN框架中实现,且对耦合关系强的情形,其稳健性更好,计算效率更高.
针对一种二维单温模型方程的保对称有限体元离散系统,给出了一种基于简单粗空间的非重叠DDM预条件子B.该预条件子行为涉及两类与原问题自相似的子问题的求解,数值实验表明基于该预条件子的PCG法的迭代次数弱依赖于问题的规模.进一步,针对一个简化的单温模型,通过引入线性有限元辅助系统,证明了关于B的预条件系统的条件数是渐近最优的(O(log3d/h)).
针对几种局部各向异性网格下3D线弹性问题分层二次元方程的求解,通过构造能有效消除由于网格的各向异性而产生的误差高频部分的特殊块磨光算子,获得了一种收敛性基本不依赖于网格规模的两水平方法.进一步,通过利用现有的适用于各向异性网格问题的AMG (DAMG或DAMG-CG)法求解粗水平(线性元)方程,建立了相应的多水平方法,并将其应用于一个实际例子的求解.数值实验结果验证了算法的有效性和稳健性.
Algebraic multigrid (AMG) method and domain decomposition method(DDM) are two fast algorithms which are widely used for solving large-scale linear systems arising from the partial diferential equations (PDEs)discretization. For discretized systems of complicated PDEs, there are stillmany problems to be studied further for the two methods. In this disser-tation, we focus on the efcient (parallel) AMG and DDM research for twomodels with wide application background, the main work is as follows.
The performance of preconditioner B01is tested and analyzed for sometypical linear systems discretizated from2D3T equations. By introducingseveral factors which are able to describe coupling relationship of the threetemperatures and diagonally dominance of the diagonal sub-matrices, wepropose a new preconditioner B2, in which Block Diagonal-type and PCTL-type preconditioners are selected adaptively. Numerical results of the test-ing for some data extracted from LARED-S show that compared with theRSAMG-preconditioned and the B01-preconditioned GMRES methods, theB2-preconditioned GMRES is more robust and of higher efciency. Further-more, the parallel algorithm of Bp2(B2) for JASMIN data interface is alsoimplemented by using the strategy of processor-grouping. For some modelsof practical application background, numerical results show that comparedwith the BoomerAMG-preconditioned GMRES solver, our parallel solver isof better algorithm scalability and faster. To be more user-friendly, we ad-ditionally provide a serial interface for Bp2.
Two discrete schemes–mixed five-point scheme and SFVE scheme onSAMR for a2D stationary difusion problem are firstly discussed, the cor-relation between the behavior of solution function near the coarse-fine in-terface, the interpolation accuracy and the approximation of the numericalsolution is then analyzed. The SFVE scheme proves to be more general.We design a two-level (TL) method for the two schemes on SAMR, and alsoprove the uniform convergence of TL method for the SFVE scheme. Further- more, for a2D3T model, we obtain the corresponding adaptive PCTL pre-conditioner Btl2by replacing RSAMG with TL method for the sub-systems.One advantage of this preconditioner is that it is easy to be implementedin JASMIN. For the strong-coupling case, numerical results show that theBtl2-preconditioned GMRES is more robust and more efcient compared withthe RSAMG-preconditioned GMRES.
A substructuring preconditioner with simple coarse spaces is constructedto solve the preserving-symmetry finite volume element (SFVE) discretizedsystem for a two-dimensional radiative difusion equation. It needs to solvetwo kinds of subproblems which are self-similar to the original problem. Nu-merical results show that the iteration number of the corresponding PCGdepends weakly on the mesh size. Furthermore, by introducing an auxiliarylinear finite element discretized system, we prove that the condition numberof the preconditioned system is nearly optimal(O(log3dh)) for a simplifiedmodel.
The hierarchical quadratic finite element discretizations on some3Dtypical local anisotropic grids are discussed. By constructing special blocksmoothers which are able to efectively eliminate the high-frequency error dueto the anisotropy of the grids, we develop a type of two-level method whoseconvergence depends weakly on the mesh size. And then, we obtain thecorresponding multi-level method by applying the existing AMG (DAMG orDAMG-CG) methods which are applicable to problems on anisotropic gridsto the solution of the coarse level (linear element) equations. Furthermore,the resulting multi-level method is applied to a3D practical example. Thenumerical results verify the high efciency and robustness of the proposedalgorithms.
针对一种关于三温辐射扩散方程离散化系统的代数两层预条件子B01,测试并分析了相应的PGMRES法的算法效率.通过引入能刻画系统耦合强弱和单温子系统对角占优强弱的若干因子,设计了一种基于块对角型和PCTL型的自适应预条件子B2.对LARED-S程序产生的数据进行了测试,表明新算法比基于RSAMG和B01预条件子的PGMRES法更加稳健和高效.进一步针对JASMIN接口,为预条件子B2设计了基于进程分组策略的并行实现算法Bp2,通过对源于实际应用背景的数据进行测试,表明所设计的并行解法器比基于BoomerAMG预条件子的PGMRES解法器具有更好的算法可扩展性,且具有更好的运算效率.此外,还为预条件子Bp2提供了串行接口.
针对二维定常扩散问题在两层SAMR网格下的混合五点格式和SFVE格式,分析了数值解的逼近性与解函数在粗细界面附近的性态以及插值算子精度之间的关系,实验结果表明后者具有更好的普适性；同时为混合五点格式和SFVE格式设计两层网格(TL)法,并证明了后者的一致收敛性.针对二维三温辐射扩散问题,为SAMR网格下的混合五点格式和SFVE格式,设计了相应的自适应PCTL预条件子Btl2其中子系统利用TL法求解.与基于RSAMG预条件子的PGMRES法相比,基于Btl2的PGMRES法更适合于在JASMIN框架中实现,且对耦合关系强的情形,其稳健性更好,计算效率更高.
针对一种二维单温模型方程的保对称有限体元离散系统,给出了一种基于简单粗空间的非重叠DDM预条件子B.该预条件子行为涉及两类与原问题自相似的子问题的求解,数值实验表明基于该预条件子的PCG法的迭代次数弱依赖于问题的规模.进一步,针对一个简化的单温模型,通过引入线性有限元辅助系统,证明了关于B的预条件系统的条件数是渐近最优的(O(log3d/h)).
针对几种局部各向异性网格下3D线弹性问题分层二次元方程的求解,通过构造能有效消除由于网格的各向异性而产生的误差高频部分的特殊块磨光算子,获得了一种收敛性基本不依赖于网格规模的两水平方法.进一步,通过利用现有的适用于各向异性网格问题的AMG (DAMG或DAMG-CG)法求解粗水平(线性元)方程,建立了相应的多水平方法,并将其应用于一个实际例子的求解.数值实验结果验证了算法的有效性和稳健性.
Algebraic multigrid (AMG) method and domain decomposition method(DDM) are two fast algorithms which are widely used for solving large-scale linear systems arising from the partial diferential equations (PDEs)discretization. For discretized systems of complicated PDEs, there are stillmany problems to be studied further for the two methods. In this disser-tation, we focus on the efcient (parallel) AMG and DDM research for twomodels with wide application background, the main work is as follows.
The performance of preconditioner B01is tested and analyzed for sometypical linear systems discretizated from2D3T equations. By introducingseveral factors which are able to describe coupling relationship of the threetemperatures and diagonally dominance of the diagonal sub-matrices, wepropose a new preconditioner B2, in which Block Diagonal-type and PCTL-type preconditioners are selected adaptively. Numerical results of the test-ing for some data extracted from LARED-S show that compared with theRSAMG-preconditioned and the B01-preconditioned GMRES methods, theB2-preconditioned GMRES is more robust and of higher efciency. Further-more, the parallel algorithm of Bp2(B2) for JASMIN data interface is alsoimplemented by using the strategy of processor-grouping. For some modelsof practical application background, numerical results show that comparedwith the BoomerAMG-preconditioned GMRES solver, our parallel solver isof better algorithm scalability and faster. To be more user-friendly, we ad-ditionally provide a serial interface for Bp2.
Two discrete schemes–mixed five-point scheme and SFVE scheme onSAMR for a2D stationary difusion problem are firstly discussed, the cor-relation between the behavior of solution function near the coarse-fine in-terface, the interpolation accuracy and the approximation of the numericalsolution is then analyzed. The SFVE scheme proves to be more general.We design a two-level (TL) method for the two schemes on SAMR, and alsoprove the uniform convergence of TL method for the SFVE scheme. Further- more, for a2D3T model, we obtain the corresponding adaptive PCTL pre-conditioner Btl2by replacing RSAMG with TL method for the sub-systems.One advantage of this preconditioner is that it is easy to be implementedin JASMIN. For the strong-coupling case, numerical results show that theBtl2-preconditioned GMRES is more robust and more efcient compared withthe RSAMG-preconditioned GMRES.
A substructuring preconditioner with simple coarse spaces is constructedto solve the preserving-symmetry finite volume element (SFVE) discretizedsystem for a two-dimensional radiative difusion equation. It needs to solvetwo kinds of subproblems which are self-similar to the original problem. Nu-merical results show that the iteration number of the corresponding PCGdepends weakly on the mesh size. Furthermore, by introducing an auxiliarylinear finite element discretized system, we prove that the condition numberof the preconditioned system is nearly optimal(O(log3dh)) for a simplifiedmodel.
The hierarchical quadratic finite element discretizations on some3Dtypical local anisotropic grids are discussed. By constructing special blocksmoothers which are able to efectively eliminate the high-frequency error dueto the anisotropy of the grids, we develop a type of two-level method whoseconvergence depends weakly on the mesh size. And then, we obtain thecorresponding multi-level method by applying the existing AMG (DAMG orDAMG-CG) methods which are applicable to problems on anisotropic gridsto the solution of the coarse level (linear element) equations. Furthermore,the resulting multi-level method is applied to a3D practical example. Thenumerical results verify the high efciency and robustness of the proposedalgorithms.
引文
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