A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations

Abstract

Preservation of nonnengativity and boundedness in the finite element solution of Nagumotype equations with general anisotropic diffusion is studied. Linear finite elements and the backward Euler scheme are used for the spatial and temporal discretization, respectively. An explicit, an implicit, and two hybrid explicit-implicit treatments for the nonlinear reaction term are considered. Conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness. The effects of lumping of the mass matrix and the reaction term are also discussed. The analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size. Numerical examples are given to demonstrate the theoretical findings.

A finite volume scheme preserving the invariant region property for a class of semilinear parabolic equations on distorted meshes

“…The nonstandard finite difference method is also applied to obtain the positivity-preserving and IRP-preserving schemes for one-dimensional Nagumo equations in [14,15]. Li and Huang [13] introduces the positivity-preserving finite element method for solving Nagumo equations on triangular meshes, while the positivity property imposes a restriction on the time step and requires the meshes to be acute triangular. The MBP-preserving of numerical approximation for the complex-valued Ginzburg-Landau model is discussed in [3,4].…”

“…In recent years, there has been tremendous interest in developing the numerical methods for several types of semilinear parabolic equations, such as Allen‐Cahn equations [5, 9, 22], Fitzhugh‐Nagumo equations [13‐15, 24, 25], Ginzburg‐Landau models [3, 4] and so on. However, these numerical methods are mainly focused on preserving a specific invariant region for a specific type of equation.…”

“…FIGURE13 The numerical solution of the nine-point scheme for Example 4 on the random triangular meshes (U min = −2.2095E−2, U max = 2.0213).…”

A gradient Discretisation Method For Anisotropic Reaction Diffusion Models with applications to the dynamics of brain tumours

“…While there has been much theoretical study of this model as part of a general theory of reaction diffusion equations [37], it is typically very rare to solve models that capture the gross behaviour of glioma tumors in heterogeneous brain tissue based on data imaging. A number of different numerical approaches for the description of glioma tumors' heterogeneous rate of invasion and the dynamics of their highly diffusive nature (mostly without full convergence analysis) have been employed, for example, see [6,9,22,24,25,27,34,38,44]. However, few studies have so far been paid to the convergence analysis of non conforming methods for the reaction diffusion equation and its corresponding models.…”