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A numerical scheme for the quantum Boltzmann equation with stiff collision terms⋆

Francis Filbet, Jingwei Hu, Shi Jin (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010) 7625–7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann...

A parabolic system involving a quadratic gradient term related to the Boussinesq approximation.

Jesús Ildefonso Díaz, Jean-Michel Rakotoson, Paul G. Schmidt (2007)

RACSAM

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. Based on adequate notions of weak and strong...

A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation

Guanyu Zhou, Takahito Kashiwabara, Issei Oikawa (2017)

Applications of Mathematics

We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization...

A piecewise P2-nonconforming quadrilateral finite element

Imbunm Kim, Zhongxuan Luo, Zhaoliang Meng, Hyun NAM, Chunjae Park, Dongwoo Sheen (2013)

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

We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...

A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations

Irene Kyza (2011)

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

We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput. 75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using...

A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations*

Irene Kyza (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using...

A predator-prey model with combined death and competition terms

Joon Hyuk Kang, Jungho Lee (2010)

Czechoslovak Mathematical Journal

The existence of a positive solution for the generalized predator-prey model for two species Δ u + u ( a + g ( u , v ) ) = 0 in Ω , Δ v + v ( d + h ( u , v ) ) = 0 in Ω , u = v = 0 on ∂ Ω , are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

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