Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach: Recovering mass balance in any situation

We present novel H(div) and H¹ liftings of given piecewise polynomials over a hierarchy of simplicial meshes, based on a global solve on the coarsest mesh and on local solves on patches of mesh elements around vertices on subsequent mesh levels. This in particular allows to lift a given algebraic residual. In connection with approaches lifting the total residual, we show how to obtain guaranteed, fully computable, and constant-free upper and lower a posteriori bounds on the algebraic, total, and discretization errors; here we consider the model Poisson equation discretized by the conforming finite element method of arbitrary order and including an arbitrary iterative solver. We next formulate safe stopping criteria ensuring that the algebraic error does not dominate the total error. We also prove efficiency, i.e., equivalence of our upper total and algebraic estimates with the total and algebraic errors, respectively, up to a generic constant; this constant is polynomial-degree-independent for the total error. Numerical experiments illustrate sharp control of all error components and accurate prediction of their spatial distribution in several test problems, including cases where some classical estimators fail. The H(div)-liftings at the same time allow to recover mass balance for any problem, any numerical discretization, and any situation such as inexact solution of (nonlinear) algebraic systems or algorithm failure, which we believe is of independent interest. We demonstrate this mass balance recovery in a simulation of immiscible incompressible two-phase flow in porous media.