Finite element approximation of the Hardy constant
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917581711998976 |
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| author | Della Pietra, Francesco Fantuzzi, Giovanni Ignat, Liviu I. Masiello, Alba Lia Paoli, Gloria Zuazua, Enrique |
| author_facet | Della Pietra, Francesco Fantuzzi, Giovanni Ignat, Liviu I. Masiello, Alba Lia Paoli, Gloria Zuazua, Enrique |
| contents | We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_01580 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite element approximation of the Hardy constant Della Pietra, Francesco Fantuzzi, Giovanni Ignat, Liviu I. Masiello, Alba Lia Paoli, Gloria Zuazua, Enrique Numerical Analysis Analysis of PDEs 65N30, 46E35 We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally. |
| title | Finite element approximation of the Hardy constant |
| topic | Numerical Analysis Analysis of PDEs 65N30, 46E35 |
| url | https://arxiv.org/abs/2308.01580 |