Two new nonconforming quadrilateral finite element methods for the Brinkman problem

Abstract:

      In this paper, two new efficient fully nonconforming polynomial finite element methods on arbitrary convex quadrilaterals are constructed to solve the Brinkman problem. Both elements have 12 local degrees of freedom and are uniformly first-order convergent. For the first element, we carefully design a shape function space of only degree four, which is convenient for practical computation. Meanwhile, the velocity solution obtained by our second element has $O(h^2)$ convergence order in the case of Darcy limit. These new elements can be regarded as modifications of a known element to effectively improve the computational efficiency, only the shape function space is properly changed. We verify our theoretical findings by numerical examples.