Mathematical description of the method
El desarrollo de un algoritmo de elementos finitos para resolver un problema definido mediante ecuaciones diferenciales y condiciones de contorno requiere en general cuatro etapas:.
Los pasos anteriores permiten construir un problema de cálculo diferencial en un problema de álgebra lineal. Dicho problema en general se plantea sobre un espacio vectorial de dimensión no-finita, pero que puede resolverse aproximadamente encontrando una proyección sobre un subespacio de dimensión finita, y por tanto con un número finito de ecuaciones (aunque en general el número de ecuaciones será elevado típicamente de miles o incluso centenares de miles). La discretización en elementos finitos ayuda a construir un algoritmo de proyección sencillo, logrando además que la solución por el método de elementos finitos sea generalmente exacta en un conjunto finito de puntos. Estos puntos coinciden usualmente con los vértices de los elementos finitos o puntos destacados de los mismos. Para la resolución concreta del enorme sistema de ecuaciones algebraicas en general pueden usarse los métodos convencionales del álgebra lineal en espacios de dimensión finita.
En lo que sigue d es la dimensión del dominio, n el número de elementos finitos y N el número de nodos total.
weak formulation
The weak formulation of a differential equation allows us to convert a differential calculus problem formulated in terms of differential equations into a linear algebra problem posed on a Banach space, generally of non-finite dimension, but which can be approximated by a finite system of algebraic equations.
Given a linear differential equation of the form:.
Where the solution is a certain function defined on a d-dimensional domain, and a set of appropriate boundary conditions have been specified, it can be assumed that the function sought is an element of a function space or Banach space V and that equation () is equivalent to:.
Where V' is the dual space of V, the weak variational form is obtained by searching for the only solution such that:.
When the linear operator is an elliptic operator, the problem can be stated as a minimization problem on the Banach space.
Domain discretization
Given a domain with a continuous boundary in the Lipschitz sense a partition into n "finite elements", is a collection of n subdomains satisfying:.
Usually for practical convenience and simplicity of analysis, all "finite elements" have the same "form", that is, there is a reference domain and a collection of bijective functions:.
This reference domain is often also called isoparametric domain. In 2D analyzes (d = 2) the reference domain is typically taken to be an equilateral triangle or a square, while in 3D analyzes (d = 3), the reference domain is typically a tetrahedron or a hexahedron. Additionally, on each element, some special points will be considered, called nodes and which will generally include the vertices of the finite element and the additional condition will be required that two adjacent elements share the nodes on the subset, that is:.
Once the partition into finite elements is defined, a functional space of finite dimension is defined on each element, usually formed by polynomials. This functional space will serve to locally approximate the solution of the variational problem. The variational problem in its weak form is posed on a space of non-finite dimension, and therefore the function sought will be a function of said space. The problem in that exact form is computationally intractable, so in practice a finite-dimensional subspace of the original vector space will be considered. And instead of the exact solution of (), the projection of the original solution on said finite-dimensional vector subspace is calculated, that is, the following problem will be solved numerically:
Where:.
If the discretization is sufficiently fine and the finite functional space over each element is well chosen, the numerical solution obtained will approximate the original solution reasonably well. This will generally imply considering a very high number of finite elements and therefore a high-dimensional projection subspace. The error between the exact solution and the approximate solution can be limited thanks to Ceá's lemma"), which essentially states that the exact solution and the approximate solution satisfy:.
That is, the error will depend above all on how well the vector subspace associated with the discretization in fintio elements approximates the original vector space.
Shape and solution space functions
There are many ways to choose a set of functions that form a vector basis on which to approximate the exact solution of the problem. From a practical point of view it is useful to define a finite-dimensional vector space defined on the reference domain formed by all polynomials of degree equal to or less than a certain degree:.
Then, through the applications that apply the reference domain to each finite element, the vector space that will serve to approximate the solution is defined as:
When it is a linear function and the space is made up of polynomials then the restriction is also a polynomial. The vector space is a polynomial space in which the base of said space is formed by shape functions, which given the set of nodes of the reference domain are defined as:
This allows us to uniquely define shape functions on the real domain on which the problem is defined:
These functions can be extended to the entire domain, thanks to the fact that the set of subdomains or finite elements constitutes a partition of the entire domain:
The shape functions allow any function defined on the original domain to be projected onto the finite element space using the projector:.
Solving the equations
Given a base associated with a certain discretization of the domain, such as that given by the functions, the weak form of the problem (when the function is bilinear) can be written as a simple matrix equation:
Where N is the number of nodes. Grouping the terms and taking into account that v^h is arbitrary and that, therefore, the previous equation must be fulfilled for any value of said arbitrary vector, we have:
This is the common form of the system of equations of an element problem associated with a linear differential equation, not dependent on time. This last form is precisely the form () of the historical review. To numerically solve the system of equations (), which usually consists of thousands or even hundreds of thousands of equations, efficient algorithms are required that optimize the number of operations that must be performed and save memory.
In general, the computational complications that must be resolved in numerical resolution are:
To understand the need for numerical integration we need to see what form the weak form of the problem typically has, expressed in terms of the subdomains or finite elements. That weak form involves integrals of the form:.
Where:.
Error approximation
According to Ceá's lemma (), the error committed in the approximation of an exact solution using finite elements is limited by the approximation error, that is, the solution obtained through the FEM is the better the approximation. Since the approximation error depends crucially on the size of the elements, the greater their number, other factors being equal, the smaller the approximation error. Below we limit this approximation error that will limit the error of the finite element solution.
To do this we need to define the diameter of each subdomain or finite element:
h is a measure of the fineness of the discretization and is the maximum of the above values. It can be verified that the approximation error (and therefore the error of the solution using finite elements) is bounded by:.
Where:.
being a multiindex and the partial derivative of u associated with it. The norm of the space L(Ω).