Estimating the error of numerical solutions of systems of reaction-diffusion equations.

*(English)*Zbl 0998.65096
Mem. Am. Math. Soc. 696, 109 p. (2000).

The authors study the following system of differential equations
\[
\frac{\partial u}{\partial t}-\nabla\cdot(\varepsilon(u,x,t)\nabla u)\sum_j\beta_j(u,x,t)\frac{\partial u}{\partial x_j}=f(u,x,t),
\]
where \(u\in\mathbb{R}^D\) is an unknown vector, \(\varepsilon\) is a diagonal matrix with smooth nonnegative entries \(\varepsilon_i\), \(\beta_j\) are diagonal matrices with smooth entries dominated by the coefficients of \(\varepsilon\) and \(f\) is a smooth vector-valued function.

It is possible that \(d\) entries of \(\varepsilon_i\) \((0<d<D)\) equal zero; one obtains then a system of \(d\) reaction-diffusion equations and \(D-d\) ordinary differential equations.

The following nine important systems (or equations for \(D=1\)) are systematically investigated in the book as an illustration of the general approach: 1) bistable equation (known as the Chafee-Infante problem), 2) system for two species, 3) Hodgkin-Huxley equations, 4) Fitz-High-Naguno equations, 5) model for superconductivity of liquids, 6) Field-Noyes equations, 7) equations for flame propagation, 8) system for morphogenesis, 9) model for the spread of rabies in foxes.

Analysing the bistable problem the authors show the limitations of classical a priori estimates for numerical solutions of reaction-diffusion equations and propose on a posteriori error analysis taking into account regularity and stability properties of particular numerical solutions.

Firstly, for a system of algebraic equations they introduce the notions of the residual error and the stability factor and then they develop the general theory of a posteriori error estimation for numerical solutions of the differential problem in one and two space dimensions. The problem with Dirichlet boundary conditions is formulated as a variational one, so numerical methods are Galerkin ones, continuous or discontinuous.

Next they discuss the size of the residual errors and stability factors for the general problem and for problems with constant diffusion.

Numerical results for the nine models are given and many aspects of numerics of the method (for instance the cost of the numerical error estimate) are discussed. The brief important chapter 5 is devoted to a preservation of invariant rectangles under discretization of general reaction-diffusion (in particular for seven of standard models) and to an implication of this for the a posteriori error analysis and for the convergence of a numerical solution to an analytical one.

The rest of the book contains technicalities of proofs of some theorems.

It is possible that \(d\) entries of \(\varepsilon_i\) \((0<d<D)\) equal zero; one obtains then a system of \(d\) reaction-diffusion equations and \(D-d\) ordinary differential equations.

The following nine important systems (or equations for \(D=1\)) are systematically investigated in the book as an illustration of the general approach: 1) bistable equation (known as the Chafee-Infante problem), 2) system for two species, 3) Hodgkin-Huxley equations, 4) Fitz-High-Naguno equations, 5) model for superconductivity of liquids, 6) Field-Noyes equations, 7) equations for flame propagation, 8) system for morphogenesis, 9) model for the spread of rabies in foxes.

Analysing the bistable problem the authors show the limitations of classical a priori estimates for numerical solutions of reaction-diffusion equations and propose on a posteriori error analysis taking into account regularity and stability properties of particular numerical solutions.

Firstly, for a system of algebraic equations they introduce the notions of the residual error and the stability factor and then they develop the general theory of a posteriori error estimation for numerical solutions of the differential problem in one and two space dimensions. The problem with Dirichlet boundary conditions is formulated as a variational one, so numerical methods are Galerkin ones, continuous or discontinuous.

Next they discuss the size of the residual errors and stability factors for the general problem and for problems with constant diffusion.

Numerical results for the nine models are given and many aspects of numerics of the method (for instance the cost of the numerical error estimate) are discussed. The brief important chapter 5 is devoted to a preservation of invariant rectangles under discretization of general reaction-diffusion (in particular for seven of standard models) and to an implication of this for the a posteriori error analysis and for the convergence of a numerical solution to an analytical one.

The rest of the book contains technicalities of proofs of some theorems.

Reviewer: S.Burys (Kraków)

##### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

92D25 | Population dynamics (general) |

82D55 | Statistical mechanical studies of superconductors |

35K57 | Reaction-diffusion equations |