P₁-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems

Abstract

Abstract.A P 1 -nonconforming quadrilateral finite element is introduced for second-order elliptic problems in two dimensions. Unlike the usual quadrilateral nonconforming finite elements, which contain quadratic polynomials or polynomials of degree greater than 2, our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges. One of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements. An optimal rate of convergence is obtained. Also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral. An extension of the P 1 -nonconforming element to three dimensions is also given. Finally, several numerical results are reported to confirm the effective nature of the proposed new element. Key words. nonconforming finite elements, quadrilateral, elliptic problems AMS subject classifications. 65N30, 65N12, 65N15PII. S00361429024049231. Introduction. We are concerned with nonconforming finite element methods for second-order elliptic problems. Nonconforming elements have been used effectively especially in fluid and solid mechanics due to their stability. Recently, these elements have attracted increasing attention from scientists and engineers in more wide areas, as this type of element is potentially useful in parallel computing.The use of finite elements for Stokes problems, which is fundamental in fluid mechanics, usually requires the discrete Babuska-Brezzi condition (inf-sup condition) to be satisfied by the velocity and pressure variables, generally set in the mixed finite element formulation; for instance, the standard P 1 -P 0 pair for triangular decompositions or the Q 1 -P 0 pair for quadrilateral decompositions of the computational domain lead to checkerboard solutions for pressure. However, if the nonconforming elements introduced in [3,8,15,5] are used to approximate the velocity part instead of the usual P 1 or Q 1 elements, the Babuska-Brezzi condition is easily satisfied, and thus stable solutions are obtained. Nonconforming finite element methods have been proved to be effective for several parameter dependent elasticity problems in a stable fashion such that the methods converge independently of the Lamé parameters, while standard conforming methods fail to converge as the parameters tend to a locking limit; see [2,12,13].Moreover, in view of domain decomposition methods, the use of nonconforming elements facilitates the exchange of information across each subdomain and provides spectral radius estimates for the iterative domain decomposition operator [9].The nonconforming simplicial finite element space of lowest degree introduced by Crouzeix and Raviart [8] is identical to the corresponding conforming one (that

Generalized Park‐Sheen Finite Elements for Adaptivity

“…Theorem 2.2 (Park and Sheen [4]) Let T 4 be an edge-connected regular triangulation of a simply connected Lipschitz domain in R 2 into quadrilaterals with nodes N = {z 1 , . .…”

“…Park and Sheen [4] introduced a basis for nonconforming, piecewise linear finite elements (FE) on triangulations into quadrilaterals of simply connected domains. Moreover, adaptive mesh-refinement has recently been proven to be optimal for the related Crouzeix-Raviart [3] nonconforming FEM on triangles [5].…”

Analysis of conforming and nonconforming quadrilateral finite element methods for the Helmholtz equation

“…The elements employed in this analysis are the standard Q 1 conforming finite element, the DSSY nonconforming element [5] and the P 1 -nonconforming quadrilateral finite element [14]. Several aspects of comparative analyses of the above three elements for two or three dimensional problems are shown.…”

“…Specifically, the following three conforming and nonconforming element methods will be analyzed: (1) the standard Q 1 conforming element (abbreviated as the "Q 1 element"); (2) the DSSY nonconforming element introduced by Douglas et al [5] (abbreviated as the "DSSY NC" element, or the "DSSY" element) which is a modified rotated Q 1 element of Rannacher and Turek [15]; and (3) the P 1 -nonconforming quadrilateral(hexahedron) element [14] (abbreviated as the " P 1 -NC element"). Santos et al [20] and Zyserman et al [19] gave detailed dispersion analyses for solving the Helmholtz equation, and elastic and viscoelastic equations by comparing between the Q 1 conforming and the DSSY NC finite element methods.…”

“…In Section 2, the dispersion relation and model problem are stated. In Section 3 a brief review of the P 1 -NC quadrilateral element [14] is also given together its weak formulation. In Section 4, numerical dispersion relations are shown for two and three dimensional problems when the P 1 -NC quadrilateral elements are employed.…”