Fast Multilevel Algorithms Of Finite Element Discretizations For Several Kinds Of H(curl) And H(grad) PDEs

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.