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This paper presents several results concerning the vector potential which can be associated with a divergence‐free function in a bounded three‐dimensional domain. Different types of boundary conditions are given, for which the existence, uniqueness and regularity of the potential are studied. This is applied firstly to the finite element discretization of these potentials and secondly to a new formulation of incompressible viscous flow problems.

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We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H 1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L 2 estimates are also derived for the three methods. Introduction.Over the past five years, discontinuous Galerkin methods have become very widely used for solving a large range of computational fluid problems. They are preferred over more standard continuous methods because of their flexibility in approximating globally rough solutions, their local mass conservation, their possible definition on unstructured meshes, their potential for error control and mesh adaptation, and their straightforward applications to anisotropic materials. Another interesting aspect of these methods is that they can employ polynomials of degree k (P k ) on quadrilaterals, whose dimension is substantially lower than that of Q k , the standard space of functions on quadrilaterals. Similarly, they lead to smaller systems of equations than the locally conservative mixed finite element methods and do not require the solution of saddle-point problems.In this paper, we discuss three numerical algorithms for elliptic problems which employ discontinuous approximation spaces. The three methods are called the nonsymmetric interior penalty Galerkin (NIPG) method, the nonsymmetric constrained Galerkin (NCG) method, and the discontinuous Galerkin (DG) method. The three algorithms are closely related in that the underlying bilinear form for all three is the same and is nonsymmetric.In the NIPG method, we modify the bilinear form of the interior penalty Galerkin method treated by Douglas and Dupont [9], Wheeler [18], Arnold [2], and Wheeler and Darlow [19]. Here, we antisymmetrize the bilinear form by changing the sign of one term and as a consequence we require only a positive penalty, whereas the proofs in [18] assume that the penalty is bounded below by a problem-dependent constant. We choose the penalty term in such a way that we can establish both H 1 and L 2 optimal convergence rates.

Abstract. A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L 2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.