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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...
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...
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...
Asymptotic error expansions in the sense of -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...
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 -scheme with . 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...
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...
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...
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...
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...
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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