Superconvergence Analysis Of Electromagnetic Problems And The Construction Of Curl-Curl Conforming Finite Elements

The problem of electromagnetic field is a popular research topic nowadays.Its numerical simulations have important values.The classical electromagnetic field problems are generally described by partial differential equations(PDEs)including second order curl operator.It can be solved by finite element methods in H(curl)space.Among them,Nedelec element is the best choice.The computational cost of solving this kind of vector PDEs is undoubtedly high.Hence how to achieve the high efficiency and high accuracy at the same time in the electromagnetic computation has draw public attention.Superconvergence analysis provides an ideal approach to reach the goal.In recent years,PDEs including higher order curl operators have appeared in magnetohydrodynamic problems and transmussion eigenvalue problems frequently.There are a wide range of applications in many areas.Such problems are generally solved in H(curl2)space with an even higher computational cost.The lowest-order of curl-curl conforming elements on the square has 24 degrees of freedoms(DoFs)on each element,which is much larger than 4 DoFs of the lowest order Nedelec element.Therefore,the superconvergence of the curl-curl element is more significant in the high performance computation.The first object of this thesis is then to study the superconvergence properties of curl con-forming element and curl-curl conforming element in solving electromagnetic field problems.We choose a set of hierarchical vector basis functions,which are equivalent to the curl con-forming Nedelec elements and are composed of Legendre polynomials,to solve the second or-der electromagnetic field problems.The superconvergence phenomenon under two-dimensional square and three-dimensional cubic partitions are analyzed.Firstly,the superconvergence prop-erty of the interpolations of the electric field E and the magnetic field H can be obtained by the definition of interpolation operators and Legendre polynomials.Based on the orthogonality of Legendre polynomials and the integral identities,the superclose properties of E and H are achieved.Then the superconvergence of numerical solutions are gained in the sense of discrete norm.Based on the superconvergence points,we use PPR post-processing technology for the recovery of the polynomials,and finally obtain the global superconvergence of both E and H in continuous norms.To our best knowledge,this is the first electromagnetic field article to obtain global superconvergence in continuous norm using PPR technique.Further,we consider the model quad-curl problem.In order to apply the curl-curl conform-ing elements to solve the problems and ensure the existence and uniqueness of the solutions,we need to transform the original problem into a saddle point problem.Then a mixed finite element approximation scheme with the curl-curl conforming basis is proposed.Based on the Babuska-Brezzi theory,we have proved the superclose properties of the finite element solution in different norms.By Bramble-Hilbert lemma,the superconvergence properties of the solution in diffuerent norms are proved.The second purpose of the thesis is to explicitly construct the curl-curl conforming hierar-chical elements on the square and cubic meshes.By choosins the generalized Jacobi polynomials with indices of-1 and-2 and using the idea of the spectral element method,we construct the hierarchical curl-curl conforming basis on two-dimensional square and three-dimensional hexahedron elements,respectively.Compared with our former constructed basis,these basis functions have some obvious advantages.Firstly,the mass matrix and the stiffness matrix are both sparse owing to the orthogonality of the poly-nomials,which facilitates the linear solver of the resulted algebraic system.Secondly,all basis functions are explicitly expressed thus are easy to use in an arbitrarily high order approxima-tion scheme.It simplifies the numerical analysis on h-and p-version finite element methods for electromagnetic field problems.Numerical experiments show that our curl-curl conforming elements can achieve the spectral accuracy and have many superconvergence phenomena.We emphasize that the curl-curl conforming hexahedral elements in the thesis is original.