On Modifications of Continuous and Discrete Maximum Principles for Reaction-Diffusion Problems

Abstract

In this work, we present and discuss some modifications, in the form of two-sided estimation (and also for arbitrary source functions instead of usual sign-conditions), of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.

On continuous and discrete maximum principles for elliptic problems with the third boundary condition

“…If c and α are not constant, and f and g are general functions, then for computations of entries (which are sums of integrals over Ω and its boundary ∂Ω) in system (5), we should, in practice, use certain quadrature rules, and, thus, each such a case may require a separate analysis. For example, we can obviously prove validity of estimates (11) in our case if a simple quadrature rule considered in [8] is used. For more complicated situations, the corresponding analysis can be done as in [13].…”

“…Only recently in [8], the implications with sign-conditions (like in ( 4)) have been generalized to (more sharp and general) two-sided a priori error estimates (similarly to DMPs used in the parabolic case, see e.g. [6,7]) via arbitrarily given data for the reaction-diffusion problems with nonzero reactive terms.…”

“…However, it was only done for a special case of homogeneous Dirichlet boundary condition. Here, we apply the approach from [8] to a more complicated case of the third boundary condition given as in (1). The elliptic equations with the third boundary condition describe some real-life problems for example in electrical engineering (heat condiction in large transformers, etc), see e.g.…”

Molecular dynamics simulations of reactive neutral chemistry in an argon‐methane plasma

“…Unreactive Ar atoms gain and carry out to the walls the excess energy of hydrocarbon bond formation: it means that Ar gas is operating as a thermostat maintaining the temperature of the neutral species. For this system, we chose the Langevin thermostat, which does not affect the time integration, [ 44 ] but only modifies the forces to perform the thermostating. [ 45 ] It is used with a separate time integration corrector as a microcanonical ensemble to update the velocities and positions of the atoms.…”