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Summary:
The convergence of the semi-variational approximations to the solution of a mixed parabolic problem is investigated. The derivation of an estimate in $L_2$-norm follows the approach suggested by Dupont, using a parabolic regularity and a projection introduced by Bramble and Osborn. As a result, the second semi-variational approximation is found to possess the maximal possible order of accuracy in space and the fourth order in time.
References:
[1] I. Hlaváček: On a semi-variational method for parabolic equations. Aplikace matematiky 17 (1972), 5, 327-351, 18 (1973), l, 43-64.
[2] J. Douglas, Jr. T. Dupont: Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 7(1970), 4, 575-626.
[3] T. Dupont: Some $L^2$ error estimates for parabolic Galerkin methods.
[4] J. H. Bramble J. E. Osborn: Rate of convergence estimates for non-self adjoint eigenvalue approximations. MRC Report 1232, Univ. Wisconsin, 1972.
[5] J. L. Lions: Equations differentielles operationelles et problèmes aux limites. Springer-Verlag, Berlin, 1961. MR 0153974
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