An adaptive multi-level method for convection diffusion problems
An Adaptive Multi-level method for Convection Diffusion Problems
In this article we introduce an adaptive multi-level method in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of different time-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a corresponding adaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments, in particular for a convection-dominated problem.
An Age and Spatially Structured Population Model for Proteus Mirabilis Swarm-Colony Development
Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces. In this paper we investigate a mathematical model that has been shown to display these structures when solved numerically. The model consists of an ordinary differential equation coupled with a partial differential equation involving a first-order hyperbolic aging term together with nonlinear degenerate diffusion. The system is shown to admit global weak solutions.
An algorithm for the one-phase Stefan problem
An application of classical analysis.
An approximate layering method for multi-dimensional nonlinear parabolic systems of a certain type
An Approximation Technique for the Numerical Solution of a Stefan Problem.
An efficient approach for solving a class of nonlinear parabolic PDEs.
An efficient finite element method for nonlinear diffusion problems
An efficient generalization of the Rush--Larsen method for solving electro-physiology membrane equations.
An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems
This paper deals with nonlinear diffusion problems which include the Stefan problem, the porous medium equation and cross-diffusion systems. We provide a linear scheme for these nonlinear diffusion problems. The proposed numerical scheme has many advantages. Namely, the implementation is very easy and the ensuing linear algebraic systems are symmetric, which show low computational cost. Moreover, this scheme has the accuracy comparable to that of the wellstudied nonlinear schemes and make it possible...
An elementary approach to nonexistence of solutions of linear parabolic equations
This note presents an elementary approach to the nonexistence of solutions of linear parabolic initial-boundary value problems considered in the Feller test.
An enclosure generating modification of the method of discretization in time
An endpoint Littlewood-Paley inequality for BVP associated with the Laplacian on Lipschitz domains.
We prove a commutator inequality of Littlewood-Paley type between partial derivatives and functions of the Laplacian on a Lipschitz domain which gives interior energy estimates for some BVP. It can be seen as an endpoint inequality for a family of energy estimates.
An error estimate uniform in time for spectral Galerkin approximations for the equations for the motion of a chemical active fluid.
We study error estimates and their convergence rates for approximate solutions of spectral Galerkin type for the equations for the motion of a viscous chemical active fluid in a bounded domain. We find error estimates that are uniform in time and also optimal in the L2-norm and H1-norm. New estimates in the H(-1)-norm are given.
An evolutionary double-well problem
An existence criterion for -norms of higher-order derivatives of solutions to a homogeneous parabolic equation.
An existence result for partially regular weak solutions of certain abstract evolution equations, with an application to magneto-hydrodynamics.
An existence result to a strongly coupled degenerated system arising in tumor modeling.
An existence theorem for parabolic equations on with discontinuous nonlinearity.