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Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

Gabriel Acosta, Julián Fernández Bonder, Pablo Groisman, Julio Daniel Rossi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions ∂ u ∂ η = u p 11 v p 12 , ∂ v ∂ η = u p 21 v p 22 on ∂Ω x (0,T), where Ω is a bounded smooth domain in ℝ d . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1) , which is the same condition as...

Single input controllability of a simplified fluid-structure interaction model

Yuning Liu, Takéo Takahashi, Marius Tucsnak (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem....

Singular perturbation for the Dirichlet boundary control of elliptic problems

Faker Ben Belgacem, Henda El Fekih, Hejer Metoui (2003)

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

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...

Singular perturbation for the Dirichlet boundary control of elliptic problems

Faker Ben Belgacem, Henda El Fekih, Hejer Metoui (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...

Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations

Stefano Berrone (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time...

Solution of contaminant transport with adsorption in porous media by the method of characteristics

Jozef Kacur, Roger Van Keer (2001)

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

A new approximation scheme is presented for the mathematical model of convection-diffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.

Solution of contaminant transport with adsorption in porous media by the method of characteristics

Jozef Kacur, Roger Van Keer (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A new approximation scheme is presented for the mathematical model of convection-diffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.

Solution of degenerate parabolic variational inequalities with convection

Jozef Kacur, Roger Van Keer (2003)

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

Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

Solution of degenerate parabolic variational inequalities with convection

Jozef Kacur, Roger Van Keer (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

Solution of mechanical problems in fractured rock with the user-defined interface of COMSOL multiphysics

Škarydová, Ilona, Hokr, Milan (2015)

Programs and Algorithms of Numerical Mathematics

This paper presents the main concept and several key features of the user-defined interface of COMSOL Java API for the solution of mechanical problems in fractured rock. This commercial computational system based on FEM has yet to incorporate fractures in mechanical problems. Our aim is to solve a 2D mechanical problem with a fracture which is defined separately from finite-element discretization and the fracture properties are included through the constitutive laws. This will be performed based...

Solution of option pricing equations using orthogonal polynomial expansion

Falko Baustian, Kateřina Filipová, Jan Pospíšil (2021)

Applications of Mathematics

We study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial differential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare the obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of the Heston model at the boundary with vanishing volatility.

Currently displaying 21 – 40 of 126