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Differential equations 

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The finite element method (FEM) is a numerical technique for finding approximate solutions to partial differential equations (PDE) and their systems, as well as integral equations. In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other. Its practical application is often known as finite element analysis (FEA).
FEM is a special case of the more general Galerkin method with polynomial approximation functions. The solution approach is based on eliminating the spatial derivatives from the PDE. This approximates the PDE with
These equation systems are linear if the underlying PDE is linear, and vice versa. Algebraic equation systems are solved using numerical linear algebra methods. Ordinary differential equations that arise in transient problems are then numerically integrated using standard techniques such as Euler's method or the RungeKutta method.
In solving partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The finite element method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). Another example would be in Numerical weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.
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While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by A. Hrennikoff and R. Courant. Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete subdomains, usually called elements.
Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations (PDEs) that arise from the problem of torsion of a cylinder. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin.
The finite element method obtained its real impetus in the 1960s and 70s by the developments of J.H. Argyris and coworkers at the University of Stuttgart, R.W. Clough and coworkers at UC Berkeley, and O.C. Zienkiewicz and coworkers at the University of Swansea. Further impetus was provided in these years by available open source finite element software programs. NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV ^{[1]} widely available. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix ^{[2]}. The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism, heat transfer, and fluid dynamics; see O.C. Zienkiewicz, R.L.Taylor, and J.Z. Zhu ^{[3]}, and K.J. Bathe ^{[4]}.
We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. It is assumed that the reader is familiar with calculus and linear algebra.
P1 is a onedimensional problem
where is given, is an unknown function of , and is the second derivative of with respect to .
P2 is a twodimensional problem (Dirichlet problem)
where is a connected open region in the plane whose boundary is "nice" (e.g., a smooth manifold or a polygon), and and denote the second derivatives with respect to and , respectively.
The problem P1 can be solved "directly" by computing antiderivatives. However, this method of solving the boundary value problem works only when there is one spatial dimension and does not generalize to higherdimensional problems or to problems like . For this reason, we will develop the finite element method for P1 and outline its generalization to P2.
Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM.
After this second step, we have concrete formulae for a large but finite dimensional linear problem whose solution will approximately solve the original BVP. This finite dimensional problem is then implemented on a computer.
The first step is to convert P1 and P2 into their equivalent weak formulations.
If solves P1, then for any smooth function that satisfies the displacement boundary conditions, i.e. at and , we have
(1)
Conversely, if with satisfies (1) for every smooth function then one may show that this will solve P1. The proof is easier for twice continuously differentiable (mean value theorem), but may be proved in a distributional sense as well.
By using integration by parts on the righthandside of (1), we obtain
(2)
where we have used the assumption that .
If we integrate by parts using a form of Green's identities, we see that if solves P2, then for any ,
where denotes the gradient and denotes the dot product in the twodimensional plane. Once more can be turned into an inner product on a suitable space of "once differentiable" functions of that are zero on . We have also assumed that (see Sobolev spaces). Existence and uniqueness of the solution can also be shown.
We can loosely think of to be the absolutely continuous functions of that are at and (see Sobolev spaces). Such functions are (weakly) "once differentiable" and it turns out that the symmetric bilinear map then defines an inner product which turns into a Hilbert space (a detailed proof is nontrivial). On the other hand, the lefthandside is also an inner product, this time on the Lp space . An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique solving (2) and therefore P1. This solution is apriori only a member of , but using elliptic regularity, will be smooth if is.
P1 and P2 are ready to be discretized which leads to a common subproblem (3). The basic idea is to replace the infinite dimensional linear problem:
with a finite dimensional version:
where is a finite dimensional subspace of . There are many possible choices for (one possibility leads to the spectral method). However, for the finite element method we take to be a space of piecewise polynomial functions.
We take the interval , choose values of with and we define by:
where we define and . Observe that functions in are not differentiable according to the elementary definition of calculus. Indeed, if then the derivative is typically not defined at any , . However, the derivative exists at every other value of and one can use this derivative for the purpose of integration by parts.
We need to be a set of functions of . In the figure on the right, we have illustrated a triangulation of a 15 sided polygonal region in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space would consist of functions that are linear on each triangle of the chosen triangulation.
One often reads instead of in the literature. The reason is that one hopes that as the underlying triangular grid becomes finer and finer, the solution of the discrete problem (3) will in some sense converge to the solution of the original boundary value problem P2. The triangulation is then indexed by a real valued parameter which one takes to be very small. This parameter will be related to the size of the largest or average triangle in the triangulation. As we refine the triangulation, the space of piecewise linear functions must also change with , hence the notation . Since we do not perform such an analysis, we will not use this notation.
To complete the discretization, we must select a basis of . In the onedimensional case, for each control point we will choose the piecewise linear function in whose value is at and zero at every , i.e.,
for ; this basis is a shifted and scaled tent function. For the twodimensional case, we choose again one basis function per vertex of the triangulation of the planar region . The function is the unique function of whose value is at and zero at every .
Depending on the author, the word "element" in "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". Finite element method is not restricted to triangles (or tetrahedra in 3d, or higher order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3d, and so on). Higher order shapes (curvilinear elements) can be defined with polynomial and even nonpolynomial shapes (e.g. ellipse or circle).
Examples of methods that use higher degree piecewise polynomial basis functions are the hpFEM and spectral FEM.
More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the 'exact' solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:
The primary advantage of this choice of basis is that the inner products
and
will be zero for almost all . (The matrix containing in the location is known as the Gramian matrix.) In the one dimensional case, the support of is the interval . Hence, the integrands of and are identically zero whenever .
Similarly, in the planar case, if and do not share an edge of the triangulation, then the integrals
and
are both zero.
If we write and then problem (3), taking for , becomes
If we denote by and the column vectors and , and if we let
and
be matrices whose entries are
and
then we may rephrase (4) as
It is not, in fact, necessary to assume . For a general function , problem (3) with for becomes actually simpler, since no matrix is used,
where and for .
As we have discussed before, most of the entries of and are zero because the basis functions have small support. So we now have to solve a linear system in the unknown where most of the entries of the matrix , which we need to invert, are zero.
Such matrices are known as sparse matrices, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, is symmetric and positive definite, so a technique such as the conjugate gradient method is favored. For problems that are not too large, sparse LU decompositions and Cholesky decompositions still work well. For instance, Matlab's backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices.
The matrix is usually referred to as the stiffness matrix, while the matrix is dubbed the mass matrix.
In general, the finite element method is characterized by the following process.
A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are merely continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial differential equations, one must use smoother basis functions. For instance, for a fourth order problem such as , one may use piecewise quadratic basis functions that are .
Another consideration is the relation of the finite dimensional space to its infinite dimensional counterpart, in the examples above . A conforming element method is one in which the space is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite dimensional space is not a subspace of the original .
Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an hmethod (h is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid is bounded above by , for some and , then one has an order p method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order method will have an error of order .
If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a pmethod. If one combines these two refinement types, one obtains an hpmethod (hpFEM). In the hpFEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods (SFEM). These are not to be confused with spectral methods.
For vector partial differential equations, the basis functions may take values in .
The Applied Element Method, or AEM combines features of both FEM and Discrete element method, or (DEM).
The Generalized Finite Element Method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a partition of unity is used to “bond” these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with microscales, and problems with boundary layers.^{[5]}
The hpFEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates.^{[6]}
The hpkFEM combines adaptively, elements with variable size h, polynomial degree of the local approximations p and global differentiability of the local approximations (k1) in order to achieve best convergence rates.
This section does not cite any references or sources. (November 2010) 
The finite difference method (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:
Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics of structures) while computational fluid dynamics (CFD) tends to use FDM or other methods like finite volume method (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.
A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.
FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.
This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.^{[7]} The introduction of FEM has substantially decreased the time to take products from concept to the production line.^{[7]} It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated.^{[8]} In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.^{[7]}
FEA has also been proposed to use in stochastic modelling, for numerically solving probability models. See the references list.^{[9]}^{[10]}
