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

Papež J, Rüde U, Vohralík M, Wohlmuth B (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Computer Methods in Applied Mechanics and Engineering Elsevier

Book Volume: 371

Article Number: 113243

DOI: 10.1016/j.cma.2020.113243

Abstract

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.

Authors with CRIS profile

Ulrich Rüde Lehrstuhl für Informatik 10 (Systemsimulation)

Involved external institutions

Inria FR France (FR) Technische Universität München (TUM) DE Germany (DE)

How to cite

APA:

Papež, J., Rüde, U., Vohralík, M., & Wohlmuth, B. (2020). Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach: Recovering mass balance in any situation. Computer Methods in Applied Mechanics and Engineering, 371. https://dx.doi.org/10.1016/j.cma.2020.113243

MLA:

Papež, Jan, et al. "Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach: Recovering mass balance in any situation." Computer Methods in Applied Mechanics and Engineering 371 (2020).

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