Numerical solution of the 1 + 2 dimensional Fisherʹs equation by finite elements and the Galerkin method

In Fisherʹs equation, the mechanism of logistic growth and linear diffusion are combined in order to model the spreading and proliferation of population, e.g., in ecological contexts. A Galerkin Finite Element method in two space dimensions is presented, which discretises a 1 + 2 dimensional version of this partial differential equation, and thus, providing a system of ordinary differential equations (ODEs) whose numerical solutions approximate those of the Fisher equation. By using a particular type of form functions, the off-diagonal elements of the matrix on the left-hand side of the ODE system become negligibly small, which makes a multiplication with the inverse of this matrix unnecessary, and therefore, leads to a simpler and faster computer program with less memory and storage requirements. It can, therefore, be considered a borderline method between finite elements and finite differences. A simple growth model for coral reefs demonstrates the programʹs adaptability to practical applications.