A Priori Estimates

The functions, M satisfy the same a priori estimate as Hence,... 

Proof. We consider a parabolic regularization of the problem approximating (5.68)-(5.72). The auxiliary boundary value problem will contain two positive parameters a, 5. The first parameter is responsible for the parabolic regularization and the second one characterizes the penalty approach. Our aim is first to prove an existence of solutions for the fixed parameters a, 5 and second to justify a passage to limits as a, d —> 0. A priori estimates uniform with respect to a, 5 are needed to analyse the passage to the limits, and we shall obtain all necessary estimates while the theorem of existence is proved. 

In this case the boundary conditions (5.81) are included in (5.84). At the first step we get a priori estimates. Assume that the solutions of (5.79)-(5.82) are smooth enough. Multiply (5.79), (5.80) by Vi, Oij — ij, respectively, and integrate over fl. Taking into account that the penalty term is nonnegative this provides the inequality... 

We first obtain a priori estimates of solutions to the problem (5.121)-(5.125). 

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

Now we shall derive an additional a priori estimate which improves (5.191). It follows from (5.198) that for almost all t G (0,T)... 

The equations (5.376)-(5.379) could be considered when t = 0. In this case we see that the obtained equations with the boundary condition (5.380) exactly coincide with the elliptic boundary value problem (5.285)-(5.289). The a priori estimate of the corresponding solution ui, Wi, mi, ni is as follows,... 

Equation (8-42) can be used in the FF calculation, assuming one knows the physical properties Cl and H. Of course, it is probable that the model will contain errors (e.g., unmeasured heat losses, incorrect Cl or H). Therefore, K can be designated as an adjustable parameter that can be timed. The use of aphysical model for FF control is desirable since it provides a physical basis for the control law and gives an a priori estimate of what the timing parameters are. Note that such a model could be nonlinear [e.g., in Eq. (8-42), F and T t. re multiplied]. 

Everything just said means that in establishing convergence and in determining the order of accuracy of a scheme it is necessary to evaluate the error of approximation, discover stability and then derive estimates of the form (22) known as a priori estimates. 

We finally get one possible a priori estimate for the solution of problem (44) ... 

For problem (51) one can derive an a priori estimate of the same type as estimate (48) for problem (44). However, in this case such an estimate fails to provide with a quite reliable idea on the speed of convergence of... 

Let us obtain this a priori estimate by multiplying equation (51) by and summing over all grid nodes ofw ,. In terms of the inner products the resulting expression can be written as... 

To prove the stability of (21), we need an a priori estimate of the form (22). A derivation of some a priori estimates for the operator equation (21) will be carried out in Section 4. A difference scheme A, y = is said to be ill-posed if at least one of the conditions (l)-(2) we have imposed above is violated. 

Some a priori estimates. We now consider several simplest a priori estimates for a solution to equation (21), the form of which depends on the subsidiary information on the operator of a scheme. These estimates are typical for difference elliptic problems. 

An a priori estimate depends on the nature of the subsidiary information on the operator A. 

Negative norms. In a priori estimates (41) and (42) we were dealing with the negative norm ip i. Just for this reason the qnestion of computing negative norms for the simplest operators will be of great importance. First of all we quote a result of simple observations ... 

We have restricted ourselves to the simplest examples demonstrating how the a priori estimates that can be obtained through such an analysis for the operator equation Ay — ip apply equally well to important problems arising in theory and practice. 

These assertions follow from the representation of the approximation error in the form (6) (8) and a priori estimate (12). On the basis of the estimates for T/j and obtained in Section 3.2 we find that... 

A priori estimates. The explicit representation (8) of a solution of problem (6)-(7) in terms of Green s function lies in the background of various a priori estimates of a solution in terms of the right-hand side. It is easily seen from (8) that... 

Estimation of a solution of the Dirichlet difference problem. We make use of a priori estimates obtained in Section 2 for a grid equation of common structure for constructing a uniform estimate of a solution of the Dirichlet difference problem (24)-(26) arising in Section 1 ... 

The embedding theorem. Various a priori estimates for the equation Ay =

energy estimates imply a uniform estimate, that is, an estimate in the norm... 

It is worth noting here that on the square grid (h = h. = h) this condition is automatically fulfilled. A proper choice of (p guarantees the sixth order of accuracy of scheme (9) on any such grid. Convergence of scheme (9) with the fourth order in the space C can be established without concern of condition (11). An alternative way of covering this is to construct an a priori estimate for A z p and then apply the embedding theorem (see Section 4). 

With the aid of the above operator inequalities we are able to produce the necessary a priori estimates and justify the convergence with the rate 0(1/r ) for the scheme in hand. Observe that for p = 2 operator (16) coincides with operator (15). 

Convergence and accuracy in the space L2(wj,). We state here that the convergence of scheme (II) follows from its stability and approximation. The error z = y — u is just the solution of problem (III). Using a priori estimate (31) behind we deduce that... 

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