TY - JOUR

T1 - Approximation by quadrilateral finite elements

AU - Arnold, Douglas N.

AU - Boffi, Daniele

AU - Falk, Richard S.

N1 - Generated from Scopus record by KAUST IRTS on 2020-05-05

PY - 2002/1/1

Y1 - 2002/1/1

N2 - We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in Lp and order r in Wp1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

AB - We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in Lp and order r in Wp1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

UR - http://www.ams.org/journal-getitem?pii=S0025-5718-02-01439-4

UR - http://www.scopus.com/inward/record.url?scp=0036018011&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-02-01439-4

DO - 10.1090/S0025-5718-02-01439-4

M3 - Article

SN - 0025-5718

VL - 71

SP - 909

EP - 922

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 239

ER -