SUPERCONVERGENCE IN FINITE-ELEMENT METHODS AND MESHES THAT ARE LOCALLY SYMMETRICAL WITH RESPECT TO A POINT

Consider a second-order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution. Con sider also its numerical approximation by standard conforming finite e lement methods with, for example, fixed degree piecewise polynomials o n a quasi-uniform mesh-family (the ''h-method''). It will be shown tha t, if the finite element function spaces are locally symmetric about a point xo with respect to the antipodal map x --> x(0) - (x - x(0)), t hen superconvergence ensues at xo under mild conditions on what happen s outside a neighborhood of x(0). For piecewise polynomials of even de gree, superconvergence occurs in function values; for piecewise polyno mials of odd degree, it occurs in derivatives.