Minimal consistent finite element space for the biharmonic equation on quadrilateral grids

Abstract

In this paper, a finite element space is presented on quadrilateral grids which can provide consistent discretization for the biharmonic equations. The space consists of piecewise quadratic polynomials and is of minimal degree for the variational problem. introductionIn the study of qualitative and numerical analysis of partial differential equations and, in general, of approximation theory, we are often interested in the approximation of functions in Sobolev spaces by piecewise polynomials defined on a partition of the domain. In order for simpler interior structure, lower degree polynomials are often expected to be used. It is of theoretical and practical interests whether and how a minimal-degree finite element space can be constructed for certain problems, namely, particularly, whether and how a consistent finite element space can be constructed for H m elliptic problem with m-th degree polynomials.When the subdivision consists of simplexes, a systematic family minimal-degree family of nonconforming finite elements has been proposed by Wang and Xu [14] for 2m-th order elliptic partial differential equations in R n for any n m with polynomials with degree m. Known as Wang-Xu or Morley-Wang-Xu family, the elements have been playing bigger and bigger role in numerical analysis. The elements are constructed based on the perfect matching between the dimension of m-th degree polynomials and the dimension of (n −k)-subsimplexes with 1 k m. The generalization to the cases n < m is attracting more and more research interests, c.f., e.g., [16].When the subdivision consists of geometrical shapes other than simplex, the problem is more complicated. The minimal conforming element spaces have been constructed for H m problem on R n rectangle grids by Hu and Zhang [7], where Q k polynomials are used for 2k-th order problems. Some low-degree rectangle elements have been designed, such as the rectangular Morley element and incomplete P 3 element for biharmonic equation. It remains open if the degrees can be further reduced for consistent nonconforming element spaces with minimal degree. Generally, the cells of 2000 Mathematics Subject Classification. Primary 65L60, 65M60, 65N30, 31A30.

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

“…We also refer to [55] for relevant discussions. The approach of nonconforming spline functions has shown useful in constructing low-degree-polynomial schemes with relatively high accuracy; please refer to [41,55,56,62] for some examples for H 1 and H 2 problems. The approach is expected to be extended to cases of higher-order problems in relatively low dimensions, particularly cases not yet covered in the Morley-Wang-Xu family, and will be discussed in the future.…”

“…For higher-differentiation-order (H m , m > 1) elliptic problems, minimal-degree approximations have been studied with the lowest accuracy-order. Specifically, when the subdivision comprises simplexes, a systematic family of nonconforming finite elements has been proposed by Wang and Xu [48] for H m elliptic partial differential equations in R n for any n m with polynomials with degree m. Besides, the constructions of finite element functions that do not depend on cell-by-cell definitions can be found in [29,41,62], wherein minimal-degree finite element spaces are defined on general quadrilateral grids for H 1 and H 2 problems. In contrast, the construction of higheraccuracy-order optimal schemes is complicated for higher-differentiation-order problems, even the planar biharmonic problem, a simple fundamental model problem.…”

A lowest-degree strictly conservative finite element scheme for incompressible Stokes problem on general triangulations

“…T ψ T = 0, we are going to show all c A and c T are zero. Similar to [36], we adopt a sweeping process here. Given a ∈ X b h , let T be such that a is a vertex of T .…”

“…For the auxiliary pair V sBDFM h0 − P 1 h0 , we mainly utilize Stenberg's macroelement argument by following the procedures of [30]; then the stability of the pair V el h0 − P 0 h0 , which is a sub-pair of V sBDFM h0 − P 1 h0 , is proved just by inheriting the stability of the V sBDFM h0 − P 1 h0 . This "reduce and inherit" procedure can be found in, e.g., [36,37] where some low degree optimal schemes are designed for other problems. It can be a natural idea to generalize all technical ingredients here to other applications.…”

A low-degree strictly conservative finite element method for incompressible flows

“…This figures out a structure of discretized Stokes complex on any local macroelements. On the other hand, similar to the study of conservative pairs in [16,22] and the study of biharmonic finite elements in [15,44,50,51], the proposed global space will be embedded in a discretized Stokes complex on the whole triangulation; this will be studied in detail in future.…”