Two Methods of Galerkin Type Achieving Optimum $L^2 $ Rates of Convergence for First Order Hyperbolics

Abstract

It has been shown that the usual Galerkin procedure applied to first order hyperbolic equations does not yield the optimum L rate of convergence for all admissible finite-dimensional subspaces. Two methods are presented which overcome this difficulty.

Estimates Away From a Discontinuity for Dissipative Galerkin Methods for Hyperbolic Equations

“…This method is similar to a method first proposed by Dendy [5] and analyzed by Wahlbin, [17] and [18], who proved convergence for smooth solutions of some linear and nonlinear hyperbolic equations. This method adds dissipation to the usual method by use of a nonstandard variational principle.…”

A dissipative Galerkin method applied to some quasilinear hyperbolic equations

“…when S 11 is a space of smoothest splines, see Fix and Nassif [6], whereas, as was shown in Dupont [4], if one employs the space of Hermite cubics (^ = 4, k = 1) the error is not optimal in L 2 but one loses one power of h in accuracy. Dendy [2] has introduced a method similar to ours-his method gives L 2 optimal error estimâtes if 117(0)-W 0 \\ H i < C 2 h^, where W o is the elliptic projection of v 0 into *S*\ The idea employed in this paper of comparing the Galerkin solution to a certain projection into S 11 of the solution to (1.1) was originated in Wheeler [11] for parabolic problems, and has been used for hyperbolic problems in e.g. Dendy [2] and Dupont [5].…”

“…Dendy [2] has introduced a method similar to ours-his method gives L 2 optimal error estimâtes if 117(0)-W 0 \\ H i < C 2 h^, where W o is the elliptic projection of v 0 into *S*\ The idea employed in this paper of comparing the Galerkin solution to a certain projection into S 11 of the solution to (1.1) was originated in Wheeler [11] for parabolic problems, and has been used for hyperbolic problems in e.g. Dendy [2] and Dupont [5]. A slightly new twist is required hère in that the projection dépends on the mesh parameter h. For this reason we give the details of Section 2.…”

“…We can solve the periodic équation y + y x -hy xx = z by a Fourier series expansion, and then (2)(3)(4)(5)(6) IMIi+*IMl2<C||z||.…”

“…It is assumed that a sufficiently smooth solution exists, see Theorem 1.1, and furthermore that with a positive constant a 0 , (1. 2) a(x, U <x, t)) ^ a 0 > 0 for this solution.…”