In the paper, the H^(-1)Galerkin-collocation method with quadratures (instead of integrals) for two point boundary value problems is considered. Approximate solution is a piecewise polynomial of degree r. It is proved that the method is stable and the error in L2-norm is of order O(h^(r+1)) if the used quadrature is exact for polynomial of degree not greater than r+1.
For some variants of the finite element method there exist points having a remainder value or a derivation remainder remarkably less than those given by global norms. This phenomenon is called superconvergence and the points are called superconvergence points. The generalized problem corresponding to (1) is as follows: Let Hk(Ω) be Sobolev space and Hk0(Ω) the completion of the space C∞0(Ω) with norm ∥⋅∥k,Ω. Find u∈H10(Ω) such that for each v∈H10(Ω), (2) a(u,v)=(f,v)0 holds, where a(u,v)=∫Ω(∑n|α|=0aα(x)DαuDαv)dx, (f,v)0=∫Ωf(x)v(x)dx, Dα=Dα11⋯Dαnn,1.5pt Dαiiu=∂αiu/∂xαii, i=1,n¯¯¯¯¯¯¯¯. The approximate problem of the finite element variant considered is the following: Find uh∈Vh such that (3) for all v∈Vh, a(uh,v)=(f,v)0. The main result is the theorem: Let ai∈C(Ω¯), D1ai,D2ai∈L∞(Ω),i=1,2,∥σ∥∞L(Ω)≤σ,f∈L2(Ω). Suppose the eigenvalues of the operator L are different from zero, and u∈H4(Ω)∩H10(Ω). Then there exists h0 such that for h≤h0, h2∑P∈G|grad(u−uh)(P)|≤Ch3(|u|3+|u|4), where u and uh are the solutions of problems (2) and (3), respectively, and C is some constant independent of h. Further, |u|k={∫Ω(∑|α|=k(Dαu)2)dx}1/2, G=⋃N1N2i=1Fi(R), R={(±3√/3,±3√/3)} is a Gauss point set in the quadrant S={(ξ1,ξ2):|ξk|≤1,k=1,2}, and Fi(F(1)i,F(2)i):S→ei, ei an element; F(1)i(ξ1,ξ2)=x(i)0+h1ξ1/2, F(2)i(ξ1,ξ2)=y(i)0+h2ξ2/2, and (x(i)0,y(i)0) is the middle element.
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