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Consider finite element approximation of the Stokes equations.We present a systematic way of stabilizing it by adding bubble functions to the discrete velocity field. Another way of stabilization is also presented where the finite element spaces are kept unchanged but the discrete incompressibility condition is modified instead.

We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. L estimates for p =* 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.

Abstract. The finite element method is used to solve a second order elliptic boundary value problem on a polygonal domain. Mesh refinements and weighted Besov spaces are used to obtain optimal error estimates and inverse theorems.

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We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. L estimates for p =* 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.

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This paper analyzes mixed methods for the biharmonic problem by means of new families of mesh dependent norms which are introduced and studied. More specifically, several mixed methods are shown to be stable with respect to these norms and, as a consequence, error estimates are obtained in a simple and direct manner.