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arxiv: 2504.09637 · v2 · submitted 2025-04-13 · 🧮 math.NA · cs.NA· math.CA

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Optimal convergence rates for the finite element approximation of the Sobolev constant

Liviu I. Ignat , Enrique Zuazua
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classification 🧮 math.NA cs.NAmath.CA
keywords elementfinitesobolevapproximationconstantconvergenceoptimalrates
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We establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p- Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.

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  1. Galerkin Approximation of the Fractional Sobolev Constant

    math.NA 2026-05 unverdicted novelty 5.0

    Sharp estimates are established for the discrete optimal constant of the fractional Sobolev inequality under Galerkin approximation with piecewise linear elements on quasi-uniform meshes in the unit ball.