On the convergence of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929437472194560 |
|---|---|
| author | Eigel, Martin Hegemann, Nando |
| author_facet | Eigel, Martin Hegemann, Nando |
| contents | Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin finite element methods. This work investigates a residual based adaptive algorithm, akin to classical adaptive FEM, used to approximate the solution of the stationary diffusion equation with lognormal coefficients, i.e. with a non-affine parameter dependence of the data. It is known that the refinement procedure is reliable but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical state-of-the-art by providing a quasi-error reduction result for the adaptive solution of the lognormal stationary diffusion problem. The presented analysis generalizes previous results in that guaranteed convergence for uniformly bounded coefficients follows directly as a corollary. Moreover, it highlights the fundamental challenges with unbounded coefficients that cannot be overcome with common techniques. A computational benchmark example illustrates the main theoretical statement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02839 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the convergence of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients Eigel, Martin Hegemann, Nando Numerical Analysis 65N12, 65N15, 65N50, 65Y20, 68Q25 Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin finite element methods. This work investigates a residual based adaptive algorithm, akin to classical adaptive FEM, used to approximate the solution of the stationary diffusion equation with lognormal coefficients, i.e. with a non-affine parameter dependence of the data. It is known that the refinement procedure is reliable but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical state-of-the-art by providing a quasi-error reduction result for the adaptive solution of the lognormal stationary diffusion problem. The presented analysis generalizes previous results in that guaranteed convergence for uniformly bounded coefficients follows directly as a corollary. Moreover, it highlights the fundamental challenges with unbounded coefficients that cannot be overcome with common techniques. A computational benchmark example illustrates the main theoretical statement. |
| title | On the convergence of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients |
| topic | Numerical Analysis 65N12, 65N15, 65N50, 65Y20, 68Q25 |
| url | https://arxiv.org/abs/2302.02839 |