We study tangential vector fields on the boundary of a bounded Lipschitz domain Ω in R 3 . Our attention is focused on the definition of suitable Hilbert spaces corresponding to fractional Sobolev regularities and also on the construction of tangential differential operators on the non-smooth manifold. The theory is applied to the characterization of tangential traces for the space H(curl, Ω). Hodge decompositions are provided for the corresponding trace spaces, and an integration by parts formula is proved.

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

We consider the discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The paper is a further development of earlier work by the authors, where we treated the homogeneous equation in a Hilbert space framework. Special attention is given here to the treatment of the forcing term. The method is combined with finite-element discretization in spatial variables.

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A naturally parallelizable numerical method for approximating scalar waves in a single space variable is developed by going to a frequency domain formulation. General forms of attenuation are permitted. Convergence is established and numerical results are presented.

We consider the discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The paper is a further development of earlier work by the authors, where we treated the homogeneous equation in a Hilbert space framework. Special attention is given here to the treatment of the forcing term. The method is combined with finite-element discretization in spatial variables.

A numerical method for approximating a pseudodifferential system describing attenuated, scalar waves is introduced and analyzed. Analytic properties of the solutions of the pseudodifferential systems are determined and used to show convergence of the numerical method. Experiments using the method are reported.

SUMMARYThe objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non-conforming four-node ÿnite elements are employed. Since the convergence of the non-conforming ÿnite element is independent of the Lamà e parameters, the sti ness of the non-conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard-free property of the non-conforming element in topology optimization problems and verify it with three typical optimization examples.