Over the last few decades, mathematical models have become an increasingly important tool for Earth scientists to understand and make predictions about how our planet functions and evolves through time and space. These models consist of one or more differential equations discretised with a numerical method and solved on a computer. The most commonly used discretisation methods are the finite difference method, the finite volume method, the finite element method (FEM), and spectral methods. In theory, each method provides the same solution to the original differential equations. However, in practice, certain methods are better suited to certain problems than others. Often one method dominates within any given discipline and in the Earth Sciences, the finite difference method is the most prevalent, due to its simplicity. Although the finite element method is arguably better suited to Earth Science problems - especially those with complicated geometry and/or material behaviour - Earth Scientists have been hesitant to wholeheartedly embrace the finite element method because it is often regarded as being complicated to implement. However, this perceived difficulty reflects the fact that most textbooks on this method are written by engineers or mathematicians for engineers who have a different educational background to earth scientists and who are often interested in different applications. This is unfortunate because FEM is a remarkably flexible and powerful tool with enormous potential in the Earth Sciences that is no more difficult (or even easier) to implement than other schemes. The aim of this textbook is to provide Earth Scientists with a practical guide on how FEM can easily be used to solve a variety of Earth Science problems using the Matlab. The book will serve as a pragmatic guide to show how a variety of different Earth science problems can be translated and solved with FEM, assuming only basic programming experience. Numerous sample Matlab codes1  will be provided to illustrate how FEM is implemented in practice. The text will not deal in detail with interpretation of model results or with direct application of the model results to specific published case studies. For those who are not modelling specialists but who use existing FEM software (e.g., ABAQUS, COMSOL), the text will serve to increase the basic level of understanding and appreciation of FEM, which is considered essential to avoid misuse of black-box software and to assist in code modification and development. Text boxes will be used to provide additional details (e.g., concerning specialised Matlab usage or more advanced topics such as integration by parts, Gaussian integration) that might not be familiar to all readers. Finally, the book could be used as a textbook for an undergraduate course on mathematical modelling in Earth Science. ____________________________________1The FEM Matlab scripts are written with an emphasis on simplicity and clarity. These standalone codes could easily be modularized, optimized and transported to other more efficient languages.

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