TY - JOUR

T1 - Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach

T2 - Recovering mass balance in any situation

AU - Papež, Jan

AU - Rüde, Ulrich

AU - Vohralík, Martin

AU - Wohlmuth, Barbara

N1 - Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - 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.

AB - 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.

KW - A posteriori error estimate

KW - Finite element method

KW - Iterative algebraic solver

KW - Mass balance

KW - Porous media flow

KW - p-robustness

UR - http://www.scopus.com/inward/record.url?scp=85088240375&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2020.113243

DO - 10.1016/j.cma.2020.113243

M3 - Article

AN - SCOPUS:85088240375

SN - 0045-7825

VL - 371

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

M1 - 113243

ER -