Some Ultra-convergent Derivative Recovery Schemes For Piecewise Linear Interpolation

In this paper,we give the error expressions of piecewise linear Lagrange polynomial interpolation and piecewise cubic polynomial interpolation using the Taylor expansions. After analyzing these expressions we give two kinds of derivative recovery schemes which can approximate the derivatives on the symmetry points with fourth order. Among them one is the derivatives recovery schemes of the piecewise linear polynomial interpolation on the near units which can be used on the uniform grids, and the other is the derivatives recovery schemes of the cubic polynomial interpolation which can be used on the six pieces strongly regular grids. With the original values on the vertex we can use the schemes to get a super-approximate value of derivatives of the original function on the symmetry points, which will improve the precision to fourth order from second order of the average derivative. We give the experiments of these two schemes . The least square approximation also has superconvergence phenomenon. Additionally we give three experiments which use the least square approximation. With the superapproximation between the finite element and the interpolation of the original function, we can quickly get a superconvergence result of the derivatives easily by using the conclusion of this paper.