A Study On The Triangular Matrix Preconditioner Of The Conjugate Gradient Method For Harmonic Electromagnetic Problems

The analysis of electromagnetic scattering problems is a research subject for agreat deal of applications. In general, the problems can be modelled in two forms ofintegral or partial equations. The unknown number of a linear equations system andthe sparseness of the nonzero elements of the matrix should be taken intoconsideration together, for solving large scale problems by using numerical methods.On one hand, the methods for partial equation are superior in storage. On the otherhand, the sparseness of the obtained high-order matrix is suitable to be solved by themeans of Conjugate Gradient Method ( CGM ) to reduce the arithmetic operations periteration. However convergence of CGM will be slow a high-order matrix of theequations. To explore proper preconditioners is an important task for numericallysolving electromagnetic scattering problems with high efficiency, which has been ahighlight in computational electromagnetics in recent years.A triangular matrix preconditioner based on the difference matricescorresponding to differential operators is presented in this thesis. A decompositionidea to construct high efficient preconditioner in the process of operator discretizationis emphasized. The combinations with optional splitting preconditioner are properlydetermined. The numerical results show the validity of the preconditioner. It can beconcluded that the idea is worthy of being referred for construction of otherpreconditioners of differential equations.In addition, matrix element values corresponded to the absorb boundary nodes areproperly decomposed along the coordinates, and the new technique is proposed toimprove acceleration effect of the preconditioner and to reduce time consuming,which can also be used for unequal mesh steps in different coordinate directions. Thecomparison between incomplete LU preconditioner and the presented preconditionersare illustrated in numerical results, which verify the convergence improvement andthe applicability. The technique of coefficient decomposition of Mur absorbingboundary condition ( ABC ) equations precisely improves the preconditioning effect .It can be considered to decompose the discrete equations of other ABC or truncationboundary condition. Further more, the triangular matrix preconditioner is generalizedto the case of 3 dimension.It is concluded from the results of this thesis that, the preconditioners obtained byuse of detailed operators structures of electromagnetic equations could be moreefficient than those general preconditioner. The proper approximation of thecoefficients of discrete truncation boundary equations can produce more efficiency ofthe preconditioner. Effect and adaption of the triangular preconditioners are verifiedby works of this thesis.