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Reduced basis solver for stochastic Galerkin formulation of Darcy flow with uncertain material parameters

Béreš, Michal (2023)

Programs and Algorithms of Numerical Mathematics

In this contribution, we present a solution to the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with uncertain material coefficients in the separable form. The SG system of equations is kept in the compressed tensor form and its solution is a very challenging task. Here, we present the reduced basis (RB) method as a solver which looks for a low-rank representation of the solution. The construction of the RB consists of iterative expanding of the basis using Monte...

Reduced order controllers for Burgers' equation with a nonlinear observer

Jeanne Atwell, Jeffrey Borggaard, Belinda King (2001)

International Journal of Applied Mathematics and Computer Science

A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers' equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal...

Relating phase field and sharp interface approaches to structural topology optimization

Luise Blank, Harald Garcke, M. Hassan Farshbaf-Shaker, Vanessa Styles (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...

Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media

Shanghui Jia, Deli Li, Tang Liu, Shu Hua Zhang (2008)

Applications of Mathematics

Asymptotic error expansions in the sense of L ∞ -norm for the Raviart-Thomas mixed finite element approximation by the lowest-order rectangular element associated with a class of parabolic integro-differential equations on a rectangular domain are derived, such that the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied to increase the accuracy of the approximations for both the vector field and the scalar field by the aid of an interpolation postprocessing...

Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Stefano Berrone (2006)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ -scheme with 1 / 2 ≤ θ ≤ 1 . Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization...

Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Stefano Berrone (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization...

Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Gerd Kunert (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction–diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element...

Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Gerd Kunert (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element...

Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport*

Carlo D'Angelo, Paolo Zunino (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations, we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for...

Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport*

Carlo D'Angelo, Paolo Zunino (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations, we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for...

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