Sources/Lexikon/Finite-Elemente-Methoden

This is a copy from the book Strang 2008.

An Introduction to the Theory

The basic Ideas

The finite element method can be described in a few words. Suppose that the problem to be solved is in variational form-it may be required to find the function $u$ which minimizes a given expression of potential energy. This minimizing property leads to a differential equation for $u$ (the Euler equation), but normally an exact solution is impossible and some approximation is necessary. The Rayleigh-Ritz-Galerkin idea is to choose a finite number of trial functions ${\phi }_1,\dots {\phi }_N$ and among all their linear combinations $\sum q_j{\phi }_j$ to find the one which is minimizing. This is the Ritz approximation. The unknown weights $q_i$ are determined, not by a differential equation, but by a system of N discrete algebraic equations which the computer can handle.

The theoretical justification for this method is simple, and compelling: The minimizing process automatically seeks out the combination which is closest to $u$. Therefore, the goal is to choose trial functions ${\phi }_j$ which are convenient enough for the potential energy to be computed and minimized, and at the same time general enough to approximate closely the unknown solution $u$.

The real difficulty is the first one, to achieve convenience and computability. In theory there always exists a set of trial functions which is complete - their linear combinations fill the space of all possible solutions as $N\to \mathrm{\infty }$, and therefore the Ritz approximations converge - but to be able to compute with them is another matter. This is what finite elements have accomplished. The underlying idea is simple. It starts by a subdivision of the structure, or the region of physical interest, into smaller pieces. These pieces must be easy for the computer to record and identify; they may be triangles or rectangles. Then within each piece the trial functions are given an extremely simple form - normally they are polynomials, of at most the third or fifth degree. Boundary conditions are infinitely easier to impose locally, along the edge of a triangle or rectangle, than globally along a more complicated boundary. The accuracy of the approximation can be increased, if that is necessary, but not by the classical Ritz method of including more and more complex trial functions. Instead, the same polynomials are retained, and the subdivision is refined. The computer follows a nearly identical set of instructions, and just takes longer to finish. In fact, a large-scale finite element system can use the power of the computer, for the formulation - of approximate equations as weil as their solution, to degree never before achieved in complicated physical problems.

Unhappily none of the credit for this idea goes to numerical analysts. The method was created by structural engineers, and it was not recognized at the start as an instance of the Rayleigh-Ritz principle. The subdivision into simpler pieces, and the equations of equilibrium and compatibility between the pieces, were initially constructed on the basis of physical reasoning. The later development of more accurate elements happened in a similar way; it was recognized that increasing the degree of the polynomials would greatly improve the accuracy, but the unknowns $q_i$ computed in the discrete approximation have always retained a physica/ significance. In this respect the computer output is much easier to interpret than the weights produced by the classical method.

The whole procedure became mathematically respectable at the moment when the unknowns were identified as the coefficients in a Ritz approximation $u\approx \sum q_j{\phi }_j$, and the discrete equations were seen to be exactly the conditions for minimizing the potential energy. Surely Argyris in Germany and England, and Martin and Clough in America, were among those responsible; we dare not guess who was first. The effect was instantly to provide a sound theoretical basis for the method. As the techniques of constructing more refined elements have matured, the underlying theory has also begun to take shape.

The fundamental problem is to discover how closely piecewise polynomials can approximate an unknown solution $u$. In other words, we must determine how well finite elements - which were developed on the basis of computational simplicity - satisfy the second requirement of good trial functions, to be effective in approximation. Intuitively, any reasonable function $u$ can be approached to arbitrary accuracy by piecewise linear functions. The mathematical task is to estimate the error as closely as possible and to determine how rapidly the error decreases as the number of pieces (or the degree of the polynomial within each piece) is increased. Of course, the finite element method can proceed without the support of precise mathematical theorems; it got on pretty well for more than 10 years. But we believe it will be useful, especially in the future development of the method, to understand and consolidate what has already been done.

We have attempted a fairly complete analysis of linear problems and the displacement method. A comparable theory for fully nonlinear equations does not yet exist, although it would certainly be possible to treat nonlinear equations - in which the difficulties are confined to lower-order terms. We make a few preliminary comments on nonlinear equations, but this remains an outstanding problem for the future. In our choice of the displacement method over the alternative variational formulations described in [the book], we have opted to side with the majority. This is the most commonly used version of the finite element method. Of course, the approximation theory would be the same for all formulations, and the duality which is so rampant throughout the whole subject makes the conversion between displacement methods and force methods nearly automatic. Our goal in this chapter is to illustrate the basic steps in the finite element method:

1. The variational formulation of the problem.
2. The construction of piecewise polynomial trial functions.
3. The computation of the stiffness matrix and solution of the discrete system.
4. The estimation of accuracy in the final -Ritz approximation.

We take the opportunity, when stating the problem variationally, to insert some of the key mathematical ideas needed for a precise theory - the Hilbert spaces $H$ and their norms, the estimates for the solution in terms of the data, and the energy inner product which is naturally associated with the specific problem. With these tools, the convergence of finite elements can be proved even for a very complicated geometry. In fact, the simplicity of variational arguments permits an analysis which already goes beyond what has been achieved for finite differences.