Interpolation Error-Based A Posteriori Error Estimation for P-Refinement
Using First and Second Derivative Jumps

Hp-adaptive finite element methods require estimates of the error in the solution at the current order and one order higher.  Interpolation-error based a posteriori error estimates offer one solution to this problem.  I show how such estimates at the current order can be obtained in one dimension for odd (even) order bases by using jumps in the first (second) derivative of the finite element solution at element boundaries.  Additionally the jumps in the second (first) derivative for odd (even) order elements allow error estimates of the finite element solution one order higher than the current order to be computed.  These estimates are compared with interpolation-error based methods that use high-order derivative approximations of the finite element solution. Computational results illustrate the theory and the impact of the estimation strategy on the refinement algorithm is discussed.