Nonconforming polynomial mixed finite element for the Brinkman problem over quadrilateral meshes

High accuracy analysis of a nonconforming discrete Stokes complex over rectangular meshes

“…The 1-form and 2-form constitute a divergence-free pair for incompressible flow. We also show its higher accuracy than that derived from a usual error estimate under uniform partitions, which explains the phenomenon observed in our previous work [31]. Numerical tests verify our theoretical results.…”

“…The convergence order of the pressure can be improved by a simple postprocessing. This complex has the same key features as that provided in our previous work [31] over uniform rectangular meshes, and so a rigorous explanation for the phenomenon observed in [31] is obtained.…”

“…The unisolvency of W K with respect to T K can be found in [7]. The assertion for the second element can be directly obtained by the technique provided in Theorem 2.5 in [31] using the first element. For a given partition T h , the sets of all vertices, interior vertices, boundary vertices, edges, interior edges and boundary edges are correspondingly denoted by…”

“…It was then used as a non C 0 nonconforming element for fourth order elliptic singular perturbation problems with a uniform convergence property [8]. Very recently, we extended this method to general convex quadrilateral meshes with some slight modifications to the shape function space, and induced a divergence-free Stokes element for the Brinkman problem [31]. The associated discrete Stokes complex was also provided.…”

“…Moreover, the convergence rate in discrete H 1 -norm is one order faster than the Adini counterpart. From this element, a nonconforming discrete complex is presented using the strategy in [31]. The 1-form and 2-form constitute a divergence-free Stokes pair for incompressible flow.…”