A Reduced Basis Method for Coercive Equations with an Exact Solution Certificate and Spatio-Parameter Adaptivity: Energy-Norm and Output Error Bounds
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Bibliographic record
Abstract
We develop a reduced basis method for linear coercive parametrized partial differential equations (PDEs) with two objectives: providing an energy-norm or functional-output a posteriori error bound with respect to the exact weak solution of the PDE as opposed to the typical finite element “truth” solution; providing reliable and efficient construction of a reduced basis model through automatic adaptivity in both physical and parameter spaces. Our error bounds build on two key ingredients. The first is a minimum-residual mixed formulation which provides an approximate solution as well as an upper bound of the dual norm of the residual with respect to the infinite-dimensional function space. The second is an extension of the successive constraint method (SCM) to evaluate a lower bound of the stability constant with respect to the infinite-dimensional function space; the approach builds on a computable lower bound of the minimum eigenvalue associated with the stability constant. Both the minimum-residual mixed formulation and the extended SCM admit offline-online computational decomposition. The offline stage incorporates spatial mesh adaptation and greedy parameter sampling for both the solution approximation and the stability eigenproblem to yield a reliable online system in an efficient manner. The online stage provides an approximate solution and an a posteriori error bound with respect to the exact solution for any parameter value in complexity independent of the size of the finite element spaces. We demonstrate the effectiveness of the approach for a thermal block problem, which exhibits parameter-dependent spatial singularities.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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