Implicit Interpolation Error-Based Error Estimation for Semilinear Elliptic and Parabolic Equations in Two Space Dimensions

Several authors have proposed an error estimation strategy for the finite element method applied to linear elliptic and parabolic equations in two space dimensions based on an odd/even order dichotomy principle.   For odd-order approximations error estimates are computed directly from the finite element solution via jumps in the first derivatives across element boundaries.  With even-order approximations an error estimate is obtained by computing a second solution on each element.  Although both estimators are asymptotically exact the even-order estimators are typically more robust than the odd-order ones.  In this paper the even-order method is extended to a family of methods for all orders greater than one, thereby recovering robustness for odd-orders.  Proofs of asymptotic exactness are extended to semilinear elliptic and parabolic equations.  Computational results demonstrating the effectiveness of the approach and comparing different members of the family are presented.