The second boundary value problem is considered for two-dimensional linear and quasilinear fourth-order elliptic equations in a rectangle, when the solution belongs to classes IV~3+~ s=O,i. Using operators of exact difference schemes, schemes are constructed for which convergence-rate estimates of order O(lhl'+' ) in the mesh norm of IVz~( )) are established.
Convergence-rite estimates have often been discussed for boundary value problems for higher-order equations.
Increased demands have then been made on the smoothness of the problem, which are not satisfied in many important applied problems.
For instance, in /1,2/ schemes for fourth-order equations are constructed, which are shown to be convergent in the mesh norm of |VzZwith a rate O(lh] z) on solutions of class C (~i. These studies do not embrace IluIh,-,.(,~,= htl..v(,~,, Ilulh,-..c=~=vrai max ~ . z~tl d+j,~s s-o
Let E={x=(z=, X2): --l~<x=<~l, a=l,2} be the square in the (x,,xz) plane, and let nh be the set of all polynomials of degree k of the variables x,, z2, i.e.,
~= {p(x) : p(x)---Z a~jx,'x2'}.
We shall later need the following lemma, which is a particular case of the Bramble-Hilbert lemma /8/.