Fast Algorithms Of Edge Element Discretizations And Adaptive Finite Element Methods For Two Classes Of Maxwell Equations

Edge finite element methods are one of the most common techniques for spatial discretizations in numerically solving Maxwell equations,but the resulting discrete systems are usually large and higher ill-condition,hence constructing the corresponding fast algorithms is necessary for realistic computational electromagnetism.Meanwhile,a broad class of problems can cause strong singularities during electromagnetic field propagate.The singularity can be resolved by refine the mesh uniformly.However uniform refinement will extremely increase the number of unknowns.Adaptive methods will conquer this shortage effectually.The above researches are two very active areas today in computational electromagnetism and will be faced with many difficult problems.In this paper,we discuss the fast algorithms for edge element discretizations and adaptive finite element methods for two classes of typical Maxwell equations.The main contents of this dissertation are as follows.The fast iterative methods and efficient preconditions for high order edge discretizations of H(curl)-elliptic equations are constructed by using the stable decompositions of high order edge finite element spaces.Furthermore, we prove that both the convergent rate of iterative methods and the condition numbers of our preconditioners are independent of mesh size by strict theoretical analysis.Numerical experiments show the efficiency of the iterative methods and the robustness of the preconditioners.Quasi-optimal error estimates in both L~2-norm and H(curl)-norm for edge discretizations of the time-harmonic Maxwell's equations are obtained by using discrete Helmholtz decompositions,the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function.Furthermore, combining quasi-optimal L~2 error estimates with the corresponding interpolation error estimates,we obtain the optimal convergence order of the error function.We also construct and analysis the corresponding two-grid method. Numerical experiments confirm the theoretical results.We consider the standard Adaptive Edge Finite Element Method(AEFEM) for the H(curl)-elliptic equations with variable coefficients and the indefinite time-harmonic Maxwell's equations,respectively.As is customary in practice,AEFEM marks exclusively according to the error estimator without special treatment of oscillation and performs a minimal element refinement without the interior node property.We first prove that the AEFEM is a contraction,for the sum of the norm of error function and the scaled error estimator,between two consecutive adaptive loops.Then using this geometric decay,global lower bounds and localized upper bound of a residual type error estimate,we derive the quasi-optimal cardinality of the AEFEM.Especially for the indefinite time-harmonic Maxwell's equations,we overcome the difficulties for the indefinite property and operator curl has a large kernel. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi-optimal.