Smetana, Kathrin: A dimensional reduction approach based on the application of reduced basis methods in the context of hierarchical model reduction. 11.12.2013
Inhalt
- Introduction
- The HMR-RB approach for linear elliptic problems
- Hierarchical model reduction for linear elliptic problems
- Formulation of the reduced problem
- Example: An advection-diffusion problem
- Discretization of the reduced problem
- Basis generation with Reduced Basis Methods
- The Hierarchical Model Reduction-Reduced Basis approach
- Derivation of a parametrized 1D problem in transverse direction
- Example: An advection-diffusion problem
- Discretization of the parametrized 1D problem in transverse direction
- Basis generation with RB techniques: The Adaptive-HMR-POD algorithm
- A posteriori error estimation
- An a posteriori error estimator based on the Riesz representative of the residual
- A localized residual-type a posteriori error estimator
- Analysis of the computational costs of the HMR-RB approach
- Numerical experiments
- Application of the HMR-RB approach to nonlinear PDEs
- The HMR framework for nonlinear PDEs
- The Empirical Projection Method (EPM)
- The HMR-RB approach (using the EPM)
- Formulation of the reduced problem in the HMR-RB framework employing the EPM
- Derivation of a parametrized 1D problem in transverse direction
- The generation of parametrized 1D operator evaluations
- Example: The nonlinear diffusion equation
- Reduced and collateral basis generation with RB methods — the Adaptive-HMR-RB algorithm
- A posteriori error estimates
- An a posteriori error bound based on the Brezzi-Rapaz-Raviart Theory
- Computation of the inf-sup stability factor and the Lipschitz constant
- Analysis of the computational costs of the HMR-RB approach
- Numerical Experiments
- Approximation of skewed interfaces
- An ansatz for approximating skewed interfaces
- Exemplification for the HMR-RB approach
- Formulation of the reduced problem
- Derivation of a parametrized 1D problem in transverse direction
- Example: An advection-diffusion problem
- Numerical Experiments
- Conclusion and Perspectives
- Appendices
- Appendix to Chapter 1
- The HMR-RB approach for a domain with a curved boundary
- Formulation of the reduced problem in the Hierarchical Model Reduction framework
- Derivation of the parameter dependent one-dimensional problem in the HMR-RB approach
- Alternative derivation of the localized estimator
- Definition of the source term s of test case 2
- Appendix to Chapter 3
- Bibliography