A Class of Numerical Algorithms for Large Time Integration: The Nonlinear Galerkin Methods

On the Rate of Convergence of the Nonlinear Galerkin Methods

“…Convergence results for such schemes have been obtained by Marion and Temam [41], Jolly et al [33], and Devulder and Marion [8] for spectral bases. More general bases (finite elements, finite differences) are considered in Marion and Temam [42], Temam [52], and Chen and Temam [4] (in this regard, see also Foias and Titi [23]).…”

“…The nonlinear Galerkin method (3.4b) associated with J?x is introduced in Marion and Temam [41], while (3.4a) is studied in Devulder and Marion [8]. In both cases, the existence of a constant M depending only on (v , \f\, Xx) such that \\ym(t)\\<M for all?…”

“…> To is derived (in the case of (3.4b) the above holds provided m is large enough, depending on u0). This guarantees ( Here, the initial data uo is assumed to be given in V (see Marion and Temam [41] and Devulder and Marion [8]). …”

“…Then, Jf = graphe is an AIM of order e = KLmXm2+x. The corresponding nonlinear Galerkin method (3.4) is studied in Devulder and Marion [8], where property (3.6) is checked as well as convergence results analogous to (4.4), (4.5).…”

“…Such analysis has been carried out in Jolly et al [33] for the Kuramoto-Sivashinsky equation. In the case of (3.4b), using techniques similar to the ones in Devulder and Marion [8], one can show the existence of a constant Mo depending only on (v ,\f\,X{) and an integer m depending on (v, l/l, Xx) and Uo (through ||uo||) such that, if m> m, problem (3.4) together with ym(0) = Pmu0 possesses a unique solution ym(t) defined for all t > 0 with According to Devulder and Marion [8], for every p e 3 §m and m large enough, (4.11) possesses a unique solution (p\, q\, q) with p0x e PmV, q\ e QmV, and q G QmBv(0, Mx). We set q -Ô(/>).…”