Finite Element Analysis
The finite element method is the most popular numerical method for solving elliptic boundary value problems. In this chapter, we introduce the concept of the finite element method, the finite element interpolation theory and its application in error estimates of finite element solutions of elliptic boundary value problems. The boundary value problems considered in this chapter are linear.
From the discussion in the previous chapter, we see that the Galerkin method for a linear boundary value problem reduces to the solution of a linear system. In solving the linear system, properties of the coefficient matrix A play an essential role. For example, if the condition number of A is too big, then from a practical perspective, it is impossible to find directly an accurate solution of the system (see ). Another important issue is the sparsity of the matrix A. The matrix A is said to be sparse, if most of its entries are zero; otherwise the matrix is said to be dense. Sparseness of the matrix can be utilized for two purposes. First, the stiffness matrix is less costly to form (observing that the computation of each entry of the matrix involves a domain integration and sometimes a boundary integration as well). Second, if the coefficient matrix is sparse, then the linear system can usually be solved more efficiently. To get a sparse stiffness matrix with the Galerkin method, we use finite dimensional approximation spaces such that it is possible to choose basis functions with small support. This consideration gives rise to the idea of the finite element method, where we use piecewise (images of) smooth functions (usually polynomials) for approximations. Loosely speaking, the finite element method is a Galerkin method with the use of piecewise (images of) polynomials.
KeywordsFinite Element Analysis Element Solution Element Space Interpolation Error Regular Family
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