Finite element method

A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms, and post-processing 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 h-version, p-version, hp-version, x-FEM, 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.

We will demonstrate 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 one-dimensional problem

${\displaystyle {\mbox{ P1 }}:{\begin{cases}u''(x)=f(x){\mbox{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}}$

where ${\displaystyle f}$ is given, ${\displaystyle u}$ is an unknown function of ${\displaystyle x}$, and ${\displaystyle u''}$ is the second derivative of ${\displaystyle u}$ with respect to ${\displaystyle x}$.

P2 is a two-dimensional problem (Dirichlet problem)

${\displaystyle {\mbox{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\mbox{ in }}\Omega ,\\u=0&{\mbox{ on }}\partial \Omega ,\end{cases}}}$

where ${\displaystyle \Omega }$ is a connected open region in the ${\displaystyle (x,y)}$ plane whose boundary ${\displaystyle \partial \Omega }$ is nice (e.g., a smooth manifold or a polygon), and ${\displaystyle u_{xx}}$ and ${\displaystyle u_{yy}}$ denote the second derivatives with respect to ${\displaystyle x}$ and ${\displaystyle y}$, 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 higher-dimensional problems or problems like ${\displaystyle u+u''=f}$. 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.

• In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required for this step. The transformation is done by hand on paper.
• The second step is the discretization, where the weak form is discretized in a finite-dimensional space.

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 ${\displaystyle u}$ solves P1, then for any smooth function ${\displaystyle v}$ that satisfies the displacement boundary conditions, i.e. ${\displaystyle v=0}$ at ${\displaystyle x=0}$ and ${\displaystyle x=1}$, we have

(1) ${\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}$

Conversely, if ${\displaystyle u}$ with ${\displaystyle u(0)=u(1)=0}$ satisfies (1) for every smooth function ${\displaystyle v(x)}$ then one may show that this ${\displaystyle u}$ will solve P1. The proof is easier for twice continuously differentiable ${\displaystyle u}$ (mean value theorem), but may be proved in a distributional sense as well.

We define a new operator or map ${\displaystyle \phi (u,v)}$ by using integration by parts on the right-hand-side of (1):

(2)${\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&=\int _{0}^{1}u''(x)v(x)\,dx\\&=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}$

where we have used the assumption that ${\displaystyle v(0)=v(1)=0}$.

If we integrate by parts using a form of Green's identities, we see that if ${\displaystyle u}$ solves P2, then we may define ${\displaystyle \phi (u,v)}$ for any ${\displaystyle v}$ by

${\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}$

where ${\displaystyle \nabla }$ denotes the gradient and ${\displaystyle \cdot }$ denotes the dot product in the two-dimensional plane. Once more ${\displaystyle \,\!\phi }$ can be turned into an inner product on a suitable space ${\displaystyle H_{0}^{1}(\Omega )}$ of once differentiable functions of ${\displaystyle \Omega }$ that are zero on ${\displaystyle \partial \Omega }$. We have also assumed that ${\displaystyle v\in H_{0}^{1}(\Omega )}$ (see Sobolev spaces). Existence and uniqueness of the solution can also be shown.

We can loosely think of ${\displaystyle H_{0}^{1}(0,1)}$ to be the absolutely continuous functions of ${\displaystyle (0,1)}$ that are ${\displaystyle 0}$ at ${\displaystyle x=0}$ and ${\displaystyle x=1}$ (see Sobolev spaces). Such functions are (weakly) once differentiable and it turns out that the symmetric bilinear map ${\displaystyle \!\,\phi }$ then defines an inner product which turns ${\displaystyle H_{0}^{1}(0,1)}$ into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side ${\displaystyle \int _{0}^{1}f(x)v(x)dx}$ is also an inner product, this time on the Lp space ${\displaystyle L^{2}(0,1)}$. An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique ${\displaystyle u}$ solving (2) and therefore P1. This solution is a-priori only a member of ${\displaystyle H_{0}^{1}(0,1)}$, but using elliptic regularity, will be smooth if ${\displaystyle f}$ is.

We take the interval ${\displaystyle (0,1)}$, choose ${\displaystyle n}$ values of ${\displaystyle x}$ with ${\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}$ and we define ${\displaystyle V}$ by:

${\displaystyle V=\{v:[0,1]\rightarrow \mathbb {R} \;:v{\mbox{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\mbox{ is linear for }}k=0,\dots ,n{\mbox{, and }}v(0)=v(1)=0\}}$

where we define ${\displaystyle x_{0}=0}$ and ${\displaystyle x_{n+1}=1}$. Observe that functions in ${\displaystyle V}$ are not differentiable according to the elementary definition of calculus. Indeed, if ${\displaystyle v\in V}$ then the derivative is typically not defined at any ${\displaystyle x=x_{k}}$, ${\displaystyle k=1,\ldots ,n}$. However, the derivative exists at every other value of ${\displaystyle x}$ and one can use this derivative for the purpose of integration by parts.

A piecewise linear function in two dimensions

We need ${\displaystyle V}$ to be a set of functions of ${\displaystyle \Omega }$. In the figure on the right, we have illustrated a triangulation of a 15 sided polygonal region ${\displaystyle \Omega }$ 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 ${\displaystyle V}$ 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 real-valued parameter ${\displaystyle h>0}$ 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 ${\displaystyle V}$ must also change with ${\displaystyle h}$. For this reason, one often reads ${\displaystyle V_{h}}$ instead of ${\displaystyle V}$ in the literature. Since we do not perform such an analysis, we will not use this notation.

Interpolation of a Bessel function

16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function J₀ (black).

The linear combination of basis functions (yellow) reproduces J₀ (blue) to any desired accuracy.

To complete the discretization, we must select a basis of ${\displaystyle V}$. In the one-dimensional case, for each control point ${\displaystyle x_{k}}$ we will choose the piecewise linear function ${\displaystyle v_{k}}$ in ${\displaystyle V}$ whose value is ${\displaystyle 1}$ at ${\displaystyle x_{k}}$ and zero at every ${\displaystyle x_{j},\;j\neq k}$, i.e.,

${\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\mbox{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\mbox{ if }}x\in [x_{k},x_{k+1}],\\0&{\mbox{ otherwise}},\end{cases}}}$

for ${\displaystyle k=1,\dots ,n}$; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function ${\displaystyle v_{k}}$ per vertex ${\displaystyle x_{k}}$ of the triangulation of the planar region ${\displaystyle \Omega }$. The function ${\displaystyle v_{k}}$ is the unique function of ${\displaystyle V}$ whose value is ${\displaystyle 1}$ at ${\displaystyle x_{k}}$ and zero at every ${\displaystyle x_{j},\;j\neq k}$.

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 3-d, or higher-order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle).

Examples of methods that use higher degree piecewise polynomial basis functions are the hp-FEM 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:

• moving nodes (r-adaptivity)
• refining (and unrefined) elements (h-adaptivity)
• changing order of base functions (p-adaptivity)
• combinations of the above (hp-adaptivity).

Solving the two-dimensional problem ${\displaystyle u_{xx}+u_{yy}=-4}$ in the disk centered at the origin and radius 1, with zero boundary conditions.
(a) The triangulation.

(b) The sparse matrix L of the discretized linear system

(c) The computed solution, ${\displaystyle u(x,y)=1-x^{2}-y^{2}.}$

The primary advantage of this choice of basis is that the inner products

${\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}$

and

${\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}$

will be zero for almost all ${\displaystyle j,k}$. (The matrix containing ${\displaystyle \langle v_{j},v_{k}\rangle }$ in the ${\displaystyle (j,k)}$ location is known as the Gramian matrix.) In the one dimensional case, the support of ${\displaystyle v_{k}}$ is the interval ${\displaystyle [x_{k-1},x_{k+1}]}$. Hence, the integrands of ${\displaystyle \langle v_{j},v_{k}\rangle }$ and ${\displaystyle \phi (v_{j},v_{k})}$ are identically zero whenever ${\displaystyle |j-k|>1}$.

Similarly, in the planar case, if ${\displaystyle x_{j}}$ and ${\displaystyle x_{k}}$ do not share an edge of the triangulation, then the integrals

${\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}$

and

${\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}$

are both zero.

If we write ${\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}$ and ${\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}$ then problem (3), taking ${\displaystyle v(x)=v_{j}(x)}$ for ${\displaystyle j=1,\dots ,n}$, becomes

${\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}$ for ${\displaystyle j=1,\dots ,n.}$ (4)

If we denote by ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {f} }$ the column vectors ${\displaystyle (u_{1},\dots ,u_{n})^{t}}$ and ${\displaystyle (f_{1},\dots ,f_{n})^{t}}$, and if we let

${\displaystyle L=(L_{ij})}$

and

${\displaystyle M=(M_{ij})}$

be matrices whose entries are

${\displaystyle L_{ij}=\phi (v_{i},v_{j})}$

and

${\displaystyle M_{ij}=\int v_{i}v_{j}dx}$

then we may rephrase (4) as

${\displaystyle -L\mathbf {u} =M\mathbf {f} .}$ (5)

It is not necessary to assume ${\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}$. For a general function ${\displaystyle f(x)}$, problem (3) with ${\displaystyle v(x)=v_{j}(x)}$ for ${\displaystyle j=1,\dots ,n}$ becomes actually simpler, since no matrix ${\displaystyle M}$ is used,

${\displaystyle -L\mathbf {u} =\mathbf {b} }$, (6)

where ${\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}$ and ${\displaystyle b_{j}=\int fv_{j}dx}$ for ${\displaystyle j=1,\dots ,n}$.

As we have discussed before, most of the entries of ${\displaystyle L}$ and ${\displaystyle M}$ are zero because the basis functions ${\displaystyle v_{k}}$ have small support. So we now have to solve a linear system in the unknown ${\displaystyle \mathbf {u} }$ where most of the entries of the matrix ${\displaystyle L}$, 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, ${\displaystyle L}$ 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 ${\displaystyle L}$ is usually referred to as the stiffness matrix, while the matrix ${\displaystyle M}$ is dubbed the mass matrix.

In general, the finite element method is characterized by the following process.

• One chooses a grid for ${\displaystyle \Omega }$. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.
• Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions.

Separate consideration is the smoothness of the basis functions. For second-order elliptic boundary value problems, piecewise polynomial basis function that is 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 ${\displaystyle u_{xxxx}+u_{yyyy}=f}$, one may use piecewise quadratic basis functions that are ${\displaystyle C^{1}}$.

Another consideration is the relation of the finite-dimensional space ${\displaystyle V}$ to its infinite-dimensional counterpart, in the examples above ${\displaystyle H_{0}^{1}}$. A conforming element method is one in which space ${\displaystyle V}$ 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 ${\displaystyle H_{0}^{1}}$.

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 h-method (h is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid ${\displaystyle h}$ is bounded above by ${\displaystyle Ch^{p}}$, for some ${\displaystyle C<\infty }$ and ${\displaystyle p>0}$, then one has an order p method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order ${\displaystyle d}$ method will have an error of order ${\displaystyle p=d+1}$.

If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p-method. If one combines these two refinement types, one obtains an hp-method (hp-FEM). In the hp-FEM, 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 ${\displaystyle \mathbb {R} ^{n}}$.

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 micro-scales, and problems with boundary layers.[14]

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 hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates.[15]

The hpk-FEM combines adaptively, elements with variable size h, polynomial degree of the local approximations p and global differentiability of the local approximations (k-1) to achieve best convergence rates.

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 re-mesh 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 semi-analytical fundamental-solutionless 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.

The S-FEM, 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.

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 high-order Lagrangian interpolants and used only with certain quadrature rules.[17]

Loubignac iteration is an iterative method in finite element methods.