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We derive some regularity estimates of the solution to a time fractional diffusion equation by using the Galerkin method. The regularity estimates partially unravel the singularity structure of the solution with respect to the time variable. We show that the regularity of the weak solution can be improved by subtracting some particular forms of singular functions.

New error estimates are established for Pian and Sumihara's (PS) 4-node assumed stress hybrid quadrilateral element [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg., 20 (1984), 1685-1695], which is widely used in engineering computation. Based on an equivalent displacement-based formulation to the PS element, we show that the numerical strain and a postprocessed numerical stress are uniformly convergent with respect to the Lamé constant $λ$ on the meshes produced through the uniform bisection procedure. Within this analysis framework, we also show that both the numerical strain and stress are uniformly convergent on meshes which are stable for the $Q_1-P_0$ Stokes element.

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