Regularity

Abstract

The previous results can only guarantee the existence of weak solutions, i.e., in the case of m = 1 only first derivatives in $L^2(\it\Omega)$ can be proven. In the beginning we also asked for second derivatives satisfying the equation $\mathcal{O}(h^k)$ requires a solution in $H^{1+k}(\it\Omega)$. Therefore the crucial question is, under what conditions the weak solution also belongs to Sobolev spaces of higher order (cf. Section 9.1). Section 9.2 characterises a specific property of elliptic solutions: In the interior of the domain the solution is smoother than close to the boundary. In the case of analytic coefficients the solution is also analytic in the interior and the bounds of the (higher) derivatives improve with the distance from the boundary. This behaviour also holds for the singularity and Green’s function. In Section 9.3 the regularity properties of solutions of difference schemes is studied. When comparing the error estimates for difference methods in §4.5 with those for finite-element estimates in §8.5 one observes that the latter require much weaker smoothness of the solution. However, one gets similar estimate for difference methods if one uses suitable discrete regularity properties (cf. §9.3.3). Unfortunately, the proof of these properties is rather technical, much more involved, and inflexible compared with the finite-element case.