Bounds-constrained finite element approximation of time-dependent partial differential equations

TL;DR

This paper extends bounds-constrained finite element methods to high-order collocation Runge-Kutta schemes for time-dependent PDEs, ensuring constraints are maintained uniformly over time with demonstrated numerical examples.

Contribution

It introduces a novel reformulation of collocation schemes that guarantees bounds constraints are enforced uniformly in time for high-order methods.

Findings

Constraints are maintained in numerical simulations.

High-order accuracy is achieved in space and time.

Numerical examples validate the approach.

Abstract

Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not guarantee enforcement of bounds constraints inherent in the original problem. Previous work enforces these bounds constraints by replacing the variational equations with variational inequalities. We extend this approach to collocation-type Runge-Kutta methods for time-dependent problems, obtaining (formally) high order methods in both space and time. By using a novel reformulation of the collocation scheme, we can guarantee that the bounds constraints hold uniformly in time. Numerical examples for a model of phytoplankton growth, the heat equation, and the Cahn-Hilliard system are given.