Introduction to Finite Element Discretization
The finite element method is a flexible numerical approach for solving partial differential equations. One of the most attractive features of the method is the straightforward handling of geometrically complicated domains. It is also easy to construct hig
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Introduction to Finite Element Discretization The finite element method is a flexible numerical approach for solving partial differential equations. One of the most attractive features of the method is the straightforward handling of geometrically complicated domains. It is also easy to construct higher-order approximations. The present chapter gives an introduction to the basic ideas of finite elements and associated computational algorithms. No previous knowledge of the method is assumed. First, we present the key ideas of a discretization framework called the weighted residual method, where the finite element method arises as a special case . Particular emphasis is put on the reasoning behind the derivation of discrete equations and especially the handling of various boundary conditions. Our formulation of the discretization procedure attempts to give the reader the proper background for understanding how to operate the finite element toolbox in Diffpack. The finite element tools in Diffpack allow the user to concentrate on specifying the weighted residual statement (also referred to as the discrete weak formulation) and the essential boundary conditions. Element-by-element assembly, numerical integration over elements, etc. are automated procedures. In the present chapter we will, however, explain all details of the finite element algorithms in ID examples and show how the algorithms are coded at a fairly low level using only straightforward array manipulations. Thereby, the reader should gain a thorough understanding of how the methods work and hopefully realize how these algorithms can, at least in principle, easily be extended to treat complicated multi-dimensional PDE problems. Advanced generalized versions of the algorithms are available in Diffpack, and we focus on their usage in later chapters. After the algorithmic aspects of the finite element method are introduced, we turn to variational forms and a more precise mathematical formulation of continuous and discrete PDE problems. This framework allows derivation of generic properties of the finite element method, such as existence and uniqueness of the solution, stability estimates, best-approximation properties, error estimates, and adaptive discretizations. There are numerous textbooks on finite elements, emphasizing different aspects of the method. Some texts are written in an engineering style with special focus on structural analysis, where the method can be derived directly from physical considerations. Other texts are written in an abstract mathH P Langtangen, Computational Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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2. Introduction to Finite Element Discretization
ematical framework and emphasize the method as an optimal approach for solving certain classes of PDEs. The treatment of the finite element method in this book is mainly intuitive and informal with weight on generic algorithmic building blocks that apply to a wide range of PDEs. The emphasis on detailed hand calculations of ID problems is not only mot
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