Uniform convergence

It is clear that the Jacobian qg = 1 — of this transformation converges uniformly to the unit on as J —> 0. Introduce the notations... 

A solution of this problem can be estimated in a similar way as was done in the preceding section, but with = r i ai. In concluding this section we establish through such an analysis that scheme (59)-(61) converges uniformly at the rate 0 h ) ... 

Theorem 2 If u x) C G), that is, a solution possesses continuous derivatives in (5 = 0 + F of the first four orders, then the difference scheme converges uniformly with the rate O(h ), that is, it is of second-order accuracy, so that estimate (16) is valid. 

We give a brief survey afforded by the above results scheme (II) converges uniformly with the same rate as in the grid L2(w )-norm (see (35)) if and only if condition (39) holds. The stability condition (39) in the space C for the explicit scheme with cr = 0, namely r < coincides with the... 

Whence it follows immediately that the explicit scheme converges uniformly... 

Under such a choice of the computational algorithm the accuracy of scheme (49) will be given special investigation. We are going to show that it converges uniformly with the rate 0 h + r " ) in the case of smooth functions k x) and... 

We note in passing that is determined by formula (47), due to which scheme (49) converges uniformly with the rate 0(r - - /i ) under the aforementioned conditions on sequences of special grids. 

Alore a detailed proof of convergence of this scheme is concerned with the form (74) and a priori estimates obtained in Chapter 6, Section 2 and so it is omitted here. As a final result we deduce that scheme (70) converges uniformly with the rate 0(r -f lE). 

Having no opportunity to touch upon this topic, we refer the readers to the aforementioned chapters of the manograph The Theory oof Difference Schemes , in which the method of extraction of stationary nonhomogeneities was employed with further reference to a priori estimates of z. The forward difference scheme with cr = 1 converges uniformly with the rate 0 h + r) due to the maximum principle. 

Then scheme (21) (23) converges uniformly with the rate 0 h + r) (it is of first-order accuracy in r and of second-order accuracy in h), so that... 

As approximation schemes, wavelets trivially satisfy the Assumptions 1 and 2 of our framework. Both the Lf and the L°° error of approximation is decreased as we move to higher index spaces. More specifically, recent work (Kon and Raphael, 1993) has proved that the wavelet transform converges uniformly according to the formula... 

This guides a proper choice of the nonequidistant grid uih(Q) so that the heat source will appear at one of the nodal points. When this is the c.a.se, scheme (60) converges uniformly, on the same grounds as before, with the rate 0( h 2 + rm,T). But a special choice of the coefficients at given by the formulas of the truncated scheme with second-order accuracy (see. Chapter 3, Section 7) improves our chances of constructing the difference scheme, of accuracy. 0( /i 2 + rm,r) for any 6 [0,1], that is, disregarding to the possible locations of the heat source. 

This provides enough reason to conclude that scheme (80) converges uniformly with the rate 0(r + h2). 

We conclude that any solution w t, s) coincides with the unique periodic solution of (3.67), as soon as so t) > s, independently of the initial condition. This is a very strong stability statement of finite time convergence, uniformly on bounded subsets s. For further details see [34]. 

The function ) (E) is analytic in e and converges uniformly in E for any fixed positive e since the norm decays exponentially with n. The desired i /( ) can then be obtained as the limit... 

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