A Note on the Reliability of Goal-Oriented Error Estimates for Galerkin Finite Element Methods with Nonlinear Functionals
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911251458686976 |
|---|---|
| author | Granzow, Brian N. Bond, Stephen D. Seidl, D. Thomas Endtmayer, Bernhard |
| author_facet | Granzow, Brian N. Bond, Stephen D. Seidl, D. Thomas Endtmayer, Bernhard |
| contents | We consider estimating the discretization error in a nonlinear functional $J(u)$ in the setting of an abstract variational problem: find $u \in \mathcal{V}$ such that $B(u,φ) = L(φ) \; \forall φ\in \mathcal{V}$, as approximated by a Galerkin finite element method. Here, $\mathcal{V}$ is a Hilbert space, $B(\cdot,\cdot)$ is a bilinear form, and $L(\cdot)$ is a linear functional. We consider well-known error estimates $η$ of the form $J(u) - J(u_h) \approx η= L(z) - B(u_h, z)$, where $u_h$ denotes a finite element approximation to $u$, and $z$ denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution solution $z$. An estimate $η$ is said to be reliable if there exists a constant $C \in \mathbb{R}_{>0}$ independent of $u_h$ such that $|J(u) - J(u_h)| \leq C|η|$. We present several example pairs of bilinear forms and nonlinear functionals where reliability of $η$ is not achieved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on the Reliability of Goal-Oriented Error Estimates for Galerkin Finite Element Methods with Nonlinear Functionals Granzow, Brian N. Bond, Stephen D. Seidl, D. Thomas Endtmayer, Bernhard Numerical Analysis Computational Engineering, Finance, and Science We consider estimating the discretization error in a nonlinear functional $J(u)$ in the setting of an abstract variational problem: find $u \in \mathcal{V}$ such that $B(u,φ) = L(φ) \; \forall φ\in \mathcal{V}$, as approximated by a Galerkin finite element method. Here, $\mathcal{V}$ is a Hilbert space, $B(\cdot,\cdot)$ is a bilinear form, and $L(\cdot)$ is a linear functional. We consider well-known error estimates $η$ of the form $J(u) - J(u_h) \approx η= L(z) - B(u_h, z)$, where $u_h$ denotes a finite element approximation to $u$, and $z$ denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution solution $z$. An estimate $η$ is said to be reliable if there exists a constant $C \in \mathbb{R}_{>0}$ independent of $u_h$ such that $|J(u) - J(u_h)| \leq C|η|$. We present several example pairs of bilinear forms and nonlinear functionals where reliability of $η$ is not achieved. |
| title | A Note on the Reliability of Goal-Oriented Error Estimates for Galerkin Finite Element Methods with Nonlinear Functionals |
| topic | Numerical Analysis Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2506.09913 |