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.
Optimal convergence rates for the finite element approximation of the Sobolev constant
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abstract
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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math.NA 1years
2026 1verdicts
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Galerkin Approximation of the Fractional Sobolev Constant
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.