Element-wise a posteriori estimates based on hierarchical bases for non-linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non-linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right-hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right-hand side the residual of the p-FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non-linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non-linear parabolic cases are provided