Superconvergence in finite element methods and meshes that are locally symmetric 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. Consider also its numerical approximation by standard conforming finite element methods with, for example, fixed degree piecewise polynomials on a quasi-uniform mesh-family (the ''h-method''). It will be shown that, 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)), then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x(0). For piecewise polynomials of even degree, superconvergence occurs in function values; for piecewise polynomials of odd degree, it occurs in derivatives.