The 𝑊¹_{𝑝} stability of the Ritz projection on graded meshes

Li, Hengguang

doi:10.1090/mcom/3101

The $W^1_p$ stability of the Ritz projection on graded meshes
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Math. Comp. 86 (2017), 49-74 Request permission

Abstract:

Consider the Poisson equation on a convex polygonal domain and the finite element method of degree $m\geq 1$ associated with a family of graded meshes for possible singular solutions. We prove the stability of the Ritz projection onto the finite element space in $W^1_p$, $1<p\leq \infty$. Consequently, we obtain finite element error estimates in $W^1_p$ for $1<p\leq \infty$ and in $L^p$ for $1<p<\infty$. The key to the analysis is the use of the “index engineering” methodology in modified Kondrat$’$ev weighted spaces. We also mention possible extensions and applications of these results.

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