Some Convergence Estimates for Semidiscrete Galerkin Type Approximations for Parabolic Equations

High order accurate two-step approximations for hyperbolic equations

“…Because of (4,6), the hypotheses of theorem 2.2 are satisfied with v replaced by v+2. Hence we obtain the optimal L 2 -convergence result (2.13) with v replaced by v + 2.…”

“…Such estimâtes are conveniently stated, following [4], as L^y (h)\\Tv\\ w ,", (1.23) for some constant C, where typically for Galerkin methods y{h) = C h r , if r> 2; y(h) = Ch 2 ïnh~1,ifr = 2. For the dérivation of (1.23) we refer to [4] and the références cited therein:…”

Convergence of finite element approximation to quasilinear initial boundary value problems

“…However the nonlinearity treated by Wheeler in volves only u and not Vu. Optimal pointwise estimâtes have been established by Bramble, Schatz, Thomée and Wahlbin [2] in the linear case (u t = Au) and by Dobrowolski [5] in the quasilinear case treated by Wheeler for n = 2, in the genera! linear case for n^2.…”

Numerical approximations of algebraic Riccati equations for abstract systems modelled by analytic semigroups, and applications

“…Thus, although much of the conceptual and technical developments of the present paper are a natural generalization of the arguments in [18], there are, however, also points of departure from [18] which require a different analysis, because of the now missing property that A have compact resolvent (see Remark 5.1), which was naturally built in the parabolic problem [18]. Like [18], our treatment here uses, as a starting point, two sources: on the one hand, the properties of the continuous optimal control problem and related Algebraic Riccati Equation following the variational approach of [17]; and, on the other hand, the approximation results for analytic semigroups (see [14,15,3]). …”