Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 11

There are two ways of spreading light:

to be the candle or the mirror that reflects it.

Edith Wharton

This book presents an introduction to the Galerkin finite element method (FEM) as a powerful and general tool for approximating solution of differential equations. Our objective is twofold.

1 i) To present the main ordinary and partial differential equations (ODEs and PDEs) modeling different phenomena in science and engineering and introduce mathematical tools and environments for their analytic and numerical studies.

2 ii) To construct some common FEMs for approximate solutions of differential equations and analyze their well‐posedness (existence, uniqueness, and stability of such approximate solutions) as well as the accuracy of the approximation.

In its final step, a finite element procedure yields a linear system of equations (LSEs) where the unknowns are the approximate values of the solution at certain nodes. Then, an approximate solution is constructed by adapting piecewise polynomials of certain degree to these, approximate, nodal values.

The entries of the coefficient matrix and the right‐hand side of FEM's final LSEs consist of integrals which, e.g. for complex geometries or less smooth, and/or more complex, data, are not always easily computable. Therefore, numerical integration and quadrature rules are introduced to approximate such integrals. Furthermore, iteration procedures are included in order to efficiently compute the numerical solutions of such obtained LSEs.

Interpolation techniques are presented for both accurate polynomial approximations and also to derive basic a priori and a posteriori error estimates necessary to determine qualitative properties of the approximate solutions. That is to show how the approximate solution, in some adequate measuring environment, e.g. a certain norm, approaches the exact solution as the number of nodes, hence, the number of unknowns increases. For convenience, the frequently used classical inequalities, such as the Cauchy–Schwarz' and Poincare, likewise the inverse and trace estimates, that are of vital importance in error analysis and stability estimates, are introduced. In the theoretical abstraction, we demonstrate the fundamental solution approach based on Green's functions and prove the Riesz (Lax–Milgram) theorem which is essential in proving the existence of a unique solution for a minimization problem that in turn is equivalent both to a variational formulation as well as a corresponding boundary value problem (BVP).

Galerkin's method for solving a general differential equation is based on seeking an approximate solution, which is

1 Easy to differentiate and integrate

2 Spanned by a set of “nearly orthogonal” base functions in a finite‐dimensional vector space.

3 Satisfies Galerkin orthogonality relation.Roughly speaking, this means a closeness relation in the sense that:I). In a priori case, the difference between the exact and approximate solution is orthogonal to the finite dimensional vector space of the approximate solution.II). In a posteriori case, the residual of the approximate solution (=the difference between the left‐ and right‐hand side of an expression obtained from the differential equation where exact solution is replaced by the approximate solution) is orthogonal to the finite dimensional vector space of the approximate solution.