English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this work, we present an abstract error analysis framework for theapproximation of linear partial differential equation (PDE) problems in weakformulation. We consider approximation methods in fully discrete formulation,where the discrete and continuous spaces are possibly not embedded in a commonspace. A proper notion of consistency is designed, and, under a classicalinf-sup condition, it is shown to bound the approximation error. This errorestimate result is in the spirit of Strang's first and second lemmas, butapplicable in situations not covered by these lemmas (because of a fullydiscrete approximation space). An improved estimate is also established in aweaker norm, using the Aubin--Nitsche trick.  We then apply these abstract estimates to an anisotropic heterogeneousdiffusion model and two classical families of schemes for this model: VirtualElement and Finite Volume methods. For each of these methods, we show that theabstract results yield new error estimates with a precise and mild dependencyon the local anisotropy ratio. A key intermediate step to derive such estimatesfor Virtual Element Methods is proving optimal approximation properties of theoblique elliptic projector in weighted Sobolev seminorms. This is a resultwhose interest goes beyond the specific model and methods considered here. Wealso obtain, to our knowledge, the first clear notion of consistency for FiniteVolume methods, which leads to a generic error estimate involving the fluxesand valid for a wide range of Finite Volume schemes. An important applicationis the first error estimate for Multi-Point Flux Approximation L and G methods.  In the appendix, not included in the published version of this work, we showthat classical estimates for discontinuous Galerkin methods can be obtainedwith simplified arguments using the abstract framework.

A third Strang lemma and an Aubin-Nitsche trick for schemes in fully discrete formulation