New Mixed Finite Element Methods In H(curl;Ω) And H(div;Ω)

A coercive mixed variational formulation on H₀（curl;Ω）×H（div;Ω）ds is proposed for the generalized Maxwell problem which typically arises from computational elec-tromagnetism.The mixed variables are the electric field and a pseudo electric dis-placement field.The well-posedness of the mixed variational problem is proven in the general settings（multiply connected domain of Lipschitz-continuous boundary with a number of connected components,filling with discontinuous,anisotropic and inhomo-geneous media）;more importantly,the coercivity is established.A conforming finite element discretization is further proposed,where the electric field is approximated by H（curl;Ω）-conforming edge element while the pseudo electric displacement field by H（div;Ω）-conforming flux element.Error estimates are obtained,and in particular,the method produces an L²curl-convergent approximation and more importantly,an L²div-convergent approximation for the solution.A new div FOSLS mixed finite element method is proposed and analyzed for the first-order system in terms of the scalar variable displacement and the vectorial vari-able flux of the general second-order elliptic problems.The main feature of the method is to apply a local constant element or（bi,tri）linear element L²projection to the div term,together with a mesh-dependent div term.The method is coercive and symmetric（although the second-order elliptic problems themselves may be neither positive defi-nite nor symmetric）,allowing any combination of conforming approximations for both scalar and vectorial variables.More imporantly,the method is suitable for nonaffine quadrilateral Raviart-Thomas（RT）and nonaffine hexahedral Raviart-Thomas-Nédélec（RTN）H（div）-elements.For nonaffine quadrial RT elements in two dimensions,the proposed method provides optimal approximations for the scalar and vectorial variables for the combination Q_m-RT_lfor all m≥1,l≥0;for nonaffine hexahedral RTN el-ements in three dimensions,it provides optimal approximations for the scalar variable for the combination Q_m-RTN_mfor all m≥1 while suboptimal approximations with order m for the vectorial variable.The proposed method assumes only the same regularity as the standard Galerkin method,e.g.,the right-hand side g∈L²（Ω）.Numerical results are given to confirm the performance and the theoretical results of the new method;in addition,unexpectedly,the new method is suitable for the convection-domainated problems with inner-boundary-layers.