New mixed finite element for an elliptic problem on hexahedral grids

Abstract

In three dimensional case, none of all the known mixed finite elements have an optimal approximation order for the velocity on the hexahedral grids. In this thesis, we give conditions of what is needed for an optimal approximation order. We next propose and analyze a new family of mixed finite elements for all index $k \geq 0$ on some hexahedral grids which provides optimal approximation order for the velocity. The pressure approximation is optimal only when $k=0$. However, it is shown that we can get an optimal approximation order for the pressure $(k \geq 1)$ by a local post-processing technique. In this thesis, we consider the finite element method for solving electro-magnetics problems. In these methods, the most useful of finite element is a curl conforming element, which have continuous tangential components across adjacent elements. We introduce new curl conforming elements of higher order $(k \geq 1)$ on parallelepiped with less degrees of freedom than the existing ones. This element has smaller number of degrees of freedom than the well known Nedelec spaces, and hence it is efficient in a numerical solution. To prove error estimate for curl conforming finite element methods, we define new divergence conforming elements of higher order $(k \geq 1)$. We apply our new elements to the Maxwell`s equations.