Differential equations 

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The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.^{[1]} The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations.
Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).
The subdivision of a whole domain into simpler parts has several advantages:^{[2]}
Typical work out of the method involves:
The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer.
In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of Galerkin method. The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steadystate problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the RungeKutta method.
In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the subdomains.
The practical application of FEM is known as finite element analysis (FEA). FEA as applied in engineering is a computational tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler–Bernoulli beam equation, the heat equation, or the NavierStokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.
FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines), when the domain changes (as during a solidstate reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations.^{[citation needed]} 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 the 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.
A clear, detailed and practical presentation of this approach can be found in The Finite Element Method for Engineers.^{[3]}
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.^{[4]} Its development can be traced back to the work by A. Hrennikoff^{[5]} and R. Courant^{[6]} in the early 1940s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of the method is usually connected with name of Leonard Oganesyan.^{[7]} It was also independently rediscovered in China by Feng Kang in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the finite difference method based on variation principle. 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 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 1970s by the developments of J. H. Argyris with coworkers at the University of Stuttgart, R. W. Clough with coworkers at UC Berkeley, O. C. Zienkiewicz with coworkers Ernest Hinton, Bruce Irons^{[8]} and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with coworkers at Cornell University. Further impetus was provided in these years by available open source finite element programs. NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV^{[9]} widely available. In Norway the ship classification society Det Norske Veritas (now DNV GL) developed Sesam in 1969 for use in analysis of ships.^{[10]} A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix.^{[11]} 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.^{[12]}^{[13]}
A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms, and postprocessing procedures.
Examples of the variational formulation are the Galerkin method, the discontinuous Galerkin method, mixed methods, etc.
A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the hversion, pversion, hpversion, xFEM, isogeometric analysis, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class.
Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy.
Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. There are some very efficient postprocessors that provide for the realization of superconvergence.
The following two problems demonstrate the finite element method.
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 (BVP) works only when there is one spatial dimension and does not generalize to higherdimensional problems or 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 finitedimensional linear problem whose solution will approximately solve the original BVP. This finitedimensional 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.
We define a new operator or map by using integration by parts on the righthandside of (1):

(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 we may define for any by
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 infinitedimensional linear problem:
with a finitedimensional version:
Find such that


(3) 
where is a finitedimensional 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 hopes that as the underlying triangular mesh 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. To measure this mesh fineness, the triangulation is indexed by a realvalued 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 . For this reason, one often reads instead of in the literature. 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 the "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". The finite element method is not restricted to triangles (or tetrahedra in 3d, or higherorder simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3d, and so on). Higherorder 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 an 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
for


(4) 
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

(5) 
It is not necessary to assume . For a general function , problem (3) with for becomes actually simpler, since no matrix is used,

(6) 
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.
Separate consideration is the smoothness of the basis functions. For secondorder elliptic boundary value problems, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.) For higherorder partial differential equations, one must use smoother basis functions. For instance, for a fourthorder problem such as , one may use piecewise quadratic basis functions that are .
Another consideration is the relation of the finitedimensional space to its infinitedimensional counterpart, in the examples above . A conforming element method is one in which 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 finitedimensional 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).
Main article: Applied element method 
The AugmentedFinite Element Method is introduced by Yang and Lui whose goal was to model the weak and strong discontinuities without the need of extra DoFs as in PuM stated.
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.^{[14]}
Main article: Mixed finite element method 
The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.
The hpFEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates.^{[15]}
The hpkFEM combines adaptively, elements with variable size h, polynomial degree of the local approximations p and global differentiability of the local approximations (k1) to achieve best convergence rates.
Main article: Extended finite element method 
The extended finite element method (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space so that it can naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and remesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges.
Several research codes implement this technique to various degrees: 1. GetFEM++ 2. xfem++ 3. openxfem++
XFEM has also been implemented in codes like Altair Radios, ASTER, Morfeo, and Abaqus. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.).
The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997).^{[16]} The SBFEM has been one of the most profitable contributions in the area of numerical analysis of fracture mechanics problems. It is a semianalytical fundamentalsolutionless method which combines the advantages of both the finite element formulations and procedures and the boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.
Main article: Smoothed finite element method 
The SFEM, Smoothed Finite Element Methods, is a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method.
Main article: Spectral element method 
Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak form partial equations that are based on highorder Lagrangian interpolants and used only with certain quadrature rules.^{[17]}
Main article: Meshfree methods 
Main article: Discontinuous Galerkin method 
Main article: Finite element limit analysis 
Main article: Stretched grid method 
Loubignac iteration is an iterative method in finite element methods.
Crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters. Metals can be regarded as crystal aggregates and it behave anisotropy under deformation, for example, abnormal stress and strain localization. CPFEM based on slip (shear strain rate) can calculate dislocation, crystal orientation and other texture information to consider crystal anisotropy during the routine. Now it has been applied in the numerical study of material deformation, surface roughness, fractures and so on.
The virtual element method (VEM), introduced by Beirão da Veiga et al. (2013)^{[18]} as an extension of mimetic finite difference (MFD) methods, is a generalisation of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or polyhedra in 3D) that are highly irregular and nonconvex in shape. The name virtual derives from the fact that knowledge of the local shape function basis is not required, and is in fact never explicitly calculated.
Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of the gradient discretization method (GDM). Hence the convergence properties of the GDM, which are established for a series of problems (linear and nonlinear elliptic problems, linear, nonlinear, and degenerate parabolic problems), hold as well for these particular FEMs.
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) tend 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 the cost of the solution favors simpler, lowerorder approximation within each cell. This is especially true for 'external flow' problems, like airflow 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 the 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.^{[21]}
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. The mesh is an integral part of the model and it must be controlled carefully to give the best results. Generally the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge and further mesh refinement does not increase accuracy.^{[22]}
This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.^{[24]} The introduction of FEM has substantially decreased the time to take products from concept to the production line.^{[24]} It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated.^{[25]} 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.^{[24]}
In the 1990s FEM was proposed for use in stochastic modelling for numerically solving probability models^{[26]} and later for reliability assessment.^{[27]}