Nonconforming Galerkin methods for the Helmholtz equation

Nonconforming Galerkin methods for a Helmholtz‐like problem arising in seismology are discussed both for standard simplicial linear elements and for several new rectangular elements related to bilinear or trilinear elements. Optimal order error estimates in a broken energy norm are derived for all elements and in L² for some of the elements when proper quadrature rules are applied to the absorbing boundary condition. Domain decomposition iterative procedures are introduced for the nonconforming methods, and their convergence at a predictable rate is established. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 475–494, 2001     

A quadrilateral Morley element for biharmonic equations

In this paper, we propose a Morley-type finite element for quadrilateral meshes to solve biharmonic problems. For each quadrilateral $Q$ , the finite element space is defined by the span of $P_2(Q)$ plus two functions in $P_3(Q)$ . Each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them. The degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges. Optimal orders of convergence are proved both in discrete $H^2$ and $H^1$ seminorms. Several numerical tests confirm the convergence analysis.     

A Locking-Free Nonconforming Finite Element Method for Planar Linear Elasticity

We propose a locking-free nonconforming finite element method based on quadrilaterals to solve for the displacement variable in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is optimal and robust in the sense that the convergence estimates in the energy and L ²-norms are independent of the Lamé parameter .