Scale Of Banach Spaces Research Articles

The Rate of Convergence for the Finite Element Method

The character of the proper refinement of the elements (mesh) around the boundary is studied. It is shown that it is possible to obtain the highest (optimal) rate of convergence by this refinement.

An abstract nonlinear Cauchy-Kovalevska theorem

A nonlinear version of Ovcyannikov’s theorem is proved. If F ( u , t ) F(u,t) is an analytic function of t t real or complex and of u u varying in a scale of Banach spaces, valued in a scale of Banach spaces, the Cauchy problem u t = F ( u , t ) , u ( 0 ) = u 0 {u_t} = F(u,t),u(0) = {u_0} , has a unique analytic solution. This is an abstract version of the Cauchy-Kovalevska theorem which can be applied to equations other than partial-differential, e.g. to certain differential-convolution or, more generally, differential-pseudodifferential equations.