Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908948339097600 |
|---|---|
| author | Tran, Ngoc Tien |
| author_facet | Tran, Ngoc Tien |
| contents | This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overlineΩ) \cap W^{2,n}_\mathrm{loc}(Ω)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04202 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form Tran, Ngoc Tien Numerical Analysis 65N12, 65N15, 65N30 This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overlineΩ) \cap W^{2,n}_\mathrm{loc}(Ω)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm. |
| title | Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form |
| topic | Numerical Analysis 65N12, 65N15, 65N30 |
| url | https://arxiv.org/abs/2302.04202 |