Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra I: A Priori Estimates

ABSTRACT

Let 𝒯 _k be a sequence of triangulations of a polyhedron Ω ⊂ ℝ ⁿ and let S _k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u _k  ∈ S _k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S _k ensures optimal rates of convergence for the sequence u _k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon.

Keywords:

• Finite element approximation
• Polyhedral domain
• Regularity
• Strongly elliptic operator
• Weighted Sobolev space

Mathematics Subject Classification:

• 35J55
• 35J05
• 35D10
• 46E35

ACKNOWLEDGMENTS

The authors were partially supported by NSF grant DMS 02-09497. Constantin Bacuta was also supported by the University of Delaware Research Foundation.

Notes

¹We owe the comment on the elasticity operators to Anna Mazzucato.