Superconvergence phenomenon in the finite element method arising from averaging gradients
The elliptic boundary value problem -div(A grad u)\(=f\) in \(D\subset {\mathbb{R}}^ 2\), \(u=0\) on \(\partial D\), is solved by the finite element method. (A is an uniformly positive definite \(2\times 2\) matrix of smooth functions). The usual spaces of linear spline functions over uniform triangular meshes are used. The averaged gradient of a spline function is a piecewise linear continuous vector field, its value at any nodal point is an average of values of the gradient of the spline in the triangles incident with this nodal point. It is shown that the gradient of a smooth function can be approximated by an averaged gradient with a rate of \(O(h^ 2)\) in \(L^ 2\). This is used to prove local superconvergence for the averaged gradient of the Ritz-Galerkin approximation to the exact gradient of the solution u \((O(h^{3/2})\) in \(L^ 2\) if u is sufficiently smooth, \(O(h^ 2)\) if D is a parallelogram). It is remarked that the same results can be obtained for boundary conditions of Newton type. The given numerical examples confirm the proved theoretical accuracy.
scientific article; zbMATH DE number 3919725
scientific article