New Mixed Finite Element Methods For Maxwell's Equations And Stokes Equations

In this paper,we propose two new finite element methods for solving Maxwell equations and the computation of the eigenvalues of the Maxwell eigenproblem and a new finite element method for solving Stokes problems.These three methods are all suitable for non-H1 singular solutions and are stable,convergent and optimal.The thesis mainly includes the following three aspects:(1):New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations with singular solutions in terms of electric field and Lagrange multiplier.Nodal-continuous Lagrange elements of any order on simplexes in two-and threedimensional spaces can be used for the electric field.The multiplier is compatibly approximated always by discontinuous piecewise constant elements.For the lower-order Lagrange elements,we use composite meshes which are obtained from the elements in the master meshes with the refinements,connecting the barycentre to the vertices of each element of the master meshes,while for the higher-order Lagrange elements,the approximations for the electric field and the multiplier are defined on the same master meshes.A general theory of stability and error estimates is developed;when applied to the eigenvalue problem,we show that the proposed mixed elements provide spectralcorrect,spurious-free approximations.Essentially optimal error bounds(only up to an arbitrarily small constant)are obtained for eigenvalues and for both singular and smooth solutions.Numerical experiments are performed to illustrate the theoretical results.(2):We propose a mixed method for the computation of the eigenvalues of the Maxwell eigenproblem,in terms of the electric field and a multiplier.The method allows the Lagrange elements of any order greater than or equal to two for the electric field,while a piecewise constant element always for the multiplier.We establish inf-sup conditions which link the curl operator and the div operator through three finite element spaces(i.e.,Uh,Qh and an auxiliary finite element space Wh).With these inf-sup conditions,we construct two Fortin-type interpolations.Then,we show the well-posedness of the method,including the kernel-ellipticity and the inf-sup condition of the pair(Uh,Qh),and we obtain the uniform and optimal convergence of the discrete solution operator.By applying the Babuska-Osborn theory of mixed methods,we show that the method is spectrally-correct,spurious-free,and obtain the error bounds for eigenvalues and eigenfunctions,optimal relative to both the regularity and the order of approximation.Most importantly,we construct a type of Fortin interpolation,from which we establish and prove the key and critical discrete compactness property of the finite element method for the eigenproblem of Maxwell equations.This is the first analytical theory and result on the discrete compactness property of nodal-continuous Lagrange elements in more than 30 years.we provide numerical results of the Maxwell eigenproblem in L-shaped domain which has singular and smooth eigenfunctions to illustrate the performance of the proposed mixed method and the obtained theoretical results.(3):For Stokes problems with different boundary conditions,such as no-slip velocity Dirichlet boundary(VDB)and pressure Dirichlet boundary(PDB)in Lipschitz domain,their solutions often have different regularities.The PDB-Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to(H1(?))2,sharply in contrast to the VDB-Stokes problem whose velocity solution still belongs to(H1(?))2,and unexpectedly,some well-known inf-sup stable and convergent VDB-Stokes elements may or may no longer correctly converge.Our purpose is to develop a general theory on how the inf-sup stable and convergent elements of the VDB-Stokes problem with no-slip velocity Dirichlet boundary(VDB)are still inf-sup stable and convergent for the PDB-Stokes problem with pressure Dirichlet boundary(PDB)in Lipschitz domain.It turns out that the inf-sup condition of the PDB-Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary.In this paper we propose two families of staggered elements:staggered Taylor-Hood elements(CPl+2)d-Pl with l?1(continuous in both velocity and pressure)and staggered Fortin elements(CPm+2)d-Pmdisc with m?1(continuous in velocity and discontinuous in pressure)on triangles and tetrahedra,for solving the PDB-Stokes problem in Lipschitz domain.We show that the two families are inf-sup stable and are correctly convergent for the non-H1 singular velocity.Numerical results illustrate the proposed elements and the theoretical results.