On the Stability and Convergence of Higher-Order Mixed Finite Element Methods for Second-Order Elliptic Problems

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Abstract
We investigate the use of higher-order mixed methods for second-order elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p. Our estimates yield asymptotic convergence rates for the p- and h — p-versions of the finite element method. They also describe more accurately than previously proved estimates the increased rate of convergence expected when the h-version is used with higher-order polynomials. For our analysis, we choose the Raviart-Thomas and the Brezzi-Douglas-Marini elements and establish optimal rates of convergence in both h and p (up to arbitrary $\varepsilon > 0$).

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