Introduction to numerical methods for geologists, mathematicians and physicists

Textbook on applied numerical methods: plate tectonics, subduction, mantle convection and planetary models

Finite Difference Method (FDM) versus Finite Element Method (FEM) April 5, 2010

Filed under: numerical methods — tgerya @ 12:02 pm

This introductory textbook is based on finite difference method (FDM) which is most intuitive to understand and easy to learn for inexperienced people. In contrast, finite element method (FEM) is more difficult to study and requires stronger mathematical background and familiarity with numerical theory.

Modelling of geological processes is based on a variety of numerical methods (finite difference, finite element, finite volume, spectral etc.) among which finite difference method (FDM) and finite element method (FEM) are the most popular. It is often believed that finite element method is superior compared to other methods since it can accurately follow material interfaces. This is widely used in engineering when deformation of complex isolated objects is modelled (e.g. car crush models).

However, in geosciences in general and in geodynamics in particular advantages of using finite element method are less obvious. This is mainly related to the necessity of capturing large deformations of complex visco-elasto-plastic materials with strong lateral variations of physical properties (e.g. viscosity) and to the absence of constant pre-defined material interfaces. For example, plate boundaries and fault structures can form spontaneously in compositionally homogeneous rocks due to the strain localization processes. Generally, in geosciences finite elements and finite differences have comparable accuracy and the choice mainly depends on personal preferences, experience and background.

In contrast to finite elements that are quite complex to study and to implement finite difference methods discussed in this introductory textbook are both simple and powerful. They offer possibility to model a wide range of complex natural processes (subduction, collision, intrusions, planetary processes etc.) with numerical approaches that are easy to understand and implement in computer programs. Therefore, if you start from scratch and seek for both making rapid progress and obtaining strong numerical geodynamic modelling background – finite difference method (FDM) is a better choice for you.

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6 Responses to “Finite Difference Method (FDM) versus Finite Element Method (FEM)”

  1. Mojtaba Says:

    I request you explain for me
    What is different of FEM to FDM to FVM ?


  2. jerrin Says:

    can you explain in brief about the procedures for solving the equations of equilibrium with boundary conditions by finite element method and finite difference method…?

    • tgerya Says:

      In brief, you have to
      1) introduce a numerical grid with defined locations for all unknown parameters,
      2) discretize respective partial differential equation (or boundary condition equation) for each unknown of the grid with FDM/FEM and thus compose a system of linear equations for all unknowns,
      3) solve the system of linear equations by either iterative or direct method and thus obtain solutions for all unknowns

  3. mike Says:

    can i have some basic background and history of finite difference method pls

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