Numerical solution of the 1 + 2 dimensional Fisher's equation by finite elements and the Galerkin method

✍ Scribed by J. Roessler; H. Hüssner

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 938 KB
• Volume: 25
• Category: Article
• ISSN: 0895-7177
• DOI: 10.1016/s0895-7177(97)00015-0

✦ Synopsis

Fisher's 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 program's adaptability to practical applications.