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The finite element method (FEM) is a popular method for numerically solving
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s arising in engineering and
mathematical modeling
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
. Typical problem areas of interest include the traditional fields of
structural analysis,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
,
fluid flow, mass transport, and
electromagnetic potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. ...
.
The FEM is a general
numerical method for solving
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
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
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, exp ...
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 equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. The method approximates the unknown function over the domain.
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
Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
a phenomenon with FEM is often referred to as finite element analysis (FEA).
Basic concepts
The subdivision of a whole domain into simpler parts has several advantages:
* Accurate representation of complex geometry
* Inclusion of dissimilar material properties
* Easy representation of the total solution
* Capture of local effects.
Typical work out of the method involves:
# dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem
# systematically recombining all sets of element equations into a global system of equations for the final calculation.
The global system of equations has known solution techniques, and can be calculated from the
initial value
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ot ...
s 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 equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
. The process, in mathematical language, is to construct an integral of the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of the residual and the
weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
s 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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
* a set of
algebraic equations
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
for
steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
problems,
* a set of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s for
transient problems.
These equation sets are the element equations. They are
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using
numerical linear algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
methods, while
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
sets that arise in the transient problems are solved by numerical integration using standard techniques such as
Euler's method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit met ...
or the
Runge-Kutta 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
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. The process is often carried out by FEM software using
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
data generated from the subdomains.
The practical application of FEM is known as ''finite element analysis'' (FEA). FEA as applied in
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
is a computational tool for performing
engineering analysis Engineering analysis involves the application of scientific/mathematical analytic principles and processes to reveal the properties and state of a system, device or mechanism under study.
Engineering analysis is decompositional: it proceeds by se ...
. It includes the use of
mesh generation
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells.
Often these cells form a simplicial complex.
Usually the cells partition the geometric input domain. ...
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
such as the
Euler–Bernoulli beam equation, the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
, or the
Navier-Stokes equations expressed in either PDE or
integral equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
s, 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 solid-state 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. 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
Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in th ...
, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as
tropical cyclone
A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depend ...
s in the atmosphere, or
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid b ...
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''.
History
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
Civil may refer to:
*Civic virtue, or civility
*Civil action, or lawsuit
* Civil affairs
*Civil and political rights
*Civil disobedience
*Civil engineering
*Civil (journalism), a platform for independent journalism
*Civilian, someone not a membe ...
and
aeronautical engineering
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: Aeronautics, aeronautical engineering and Astronautics, astronautical engineering. A ...
. Its development can be traced back to the work by
A. Hrennikoff and
R. Courant in the early 1940s. Another pioneer was
Ioannis Argyris
Johann Hadji Argyris Fellow of the Royal Society, FRS (Greek alphabet, Greek: Ιωάννης Χατζι Αργύρης; 19 August 1913 – 2 April 2004) was a Greek pioneer of computer applications in science and engineering,Hughes TJR, J. Tinsley ...
. In the USSR, the introduction of the practical application of the method is usually connected with name of
Leonard Oganesyan
Leonard or ''Leo'' is a common English masculine given name and a surname.
The given name and surname originate from the Old High German ''Leonhard'' containing the prefix ''levon'' ("lion") from the Greek Λέων ("lion") through the Latin '' L ...
. 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
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, ex ...
discretization of a continuous domain into a set of discrete sub-domains, usually called elements.
Hrennikoff's work discretizes the domain by using a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
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
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of a
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
. 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 co-workers at the
University of Stuttgart
The University of Stuttgart (german: Universität Stuttgart) is a leading research university located in Stuttgart, Germany. It was founded in 1829 and is organized into 10 faculties. It is one of the oldest technical universities in Germany wit ...
,
R. W. Clough with co-workers at
UC Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public university, public land-grant university, land-grant research university in Berkeley, California. Established in 1868 as the University of Californi ...
,
O. C. Zienkiewicz with co-workers
Ernest Hinton,
Bruce Irons and others at
Swansea University
, former_names=University College of Swansea, University of Wales Swansea
, motto= cy, Gweddw crefft heb ei dawn
, mottoeng="Technical skill is bereft without culture"
, established=1920 – University College of Swansea 1996 – University of Wa ...
,
Philippe G. Ciarlet at the University of
Paris 6 and Richard Gallagher with co-workers at
Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...
. Further impetus was provided in these years by available open source finite element programs. NASA sponsored the original version of
NASTRAN
NASTRAN is a finite element analysis (FEA) program that was originally developed for NASA in the late 1960s under United States government funding for the aerospace industry. The MacNeal-Schwendler Corporation (MSC) was one of the principal and o ...
, and UC Berkeley made the finite element program SAP IV widely available. In Norway the ship classification society Det Norske Veritas (now
DNV GL
DNV (formerly DNV GL) is an international accredited registrar and classification society headquartered in Høvik, Norway. The company currently has about 12,000 employees and 350 offices operating in more than 100 countries, and provides serv ...
) developed
Sesam Sesam, SESAM or SeSaM may refer to:
* SESAM (database), a relational database developed by Fujitsu Siemens
* SESAM (FEM), a structural analysis software
* Sesam (search engine), a Scandinavian internet search engine
* SeSaM-Biotech GmbH, a biotech ...
in 1969 for use in analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by
Strang and
Fix. The method has since been generalized for the
numerical modeling of physical systems in a wide variety of
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
disciplines, e.g.,
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, and
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
.
Technical discussion
The structure of finite element methods
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
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
, 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.
Illustrative problems P1 and P2
The following two problems demonstrate the finite element method.
P1 is a one-dimensional problem
:
where
is given,
is an unknown function of
, and
is the second derivative of
with respect to
.
P2 is a two-dimensional problem (
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
)
:
where
is a connected open region in the
plane whose boundary
is nice (e.g., a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
or a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
), and
and
denote the second derivatives with respect to
and
, respectively.
The problem P1 can be solved directly by computing
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s. 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
. 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
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
.
Weak formulation
The first step is to convert P1 and P2 into their equivalent
weak formulation Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...
s.
The weak form of P1
If
solves P1, then for any smooth function
that satisfies the displacement boundary conditions, i.e.
at
and
, we have
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
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
on the right-hand-side of (1):
where we have used the assumption that
.
The weak form of P2
If we integrate by parts using a form of
Green's identities
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's ...
, we see that if
solves P2, then we may define
for any
by
:
where
denotes the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
and
denotes the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
in the two-dimensional 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 space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s). Existence and uniqueness of the solution can also be shown.
A proof outline of existence and uniqueness of the solution
We can loosely think of
to be the
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
functions of
that are
at
and
(see
Sobolev spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
). Such functions are (weakly) once differentiable and it turns out that the symmetric
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
then defines an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
which turns
into a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(a detailed proof is nontrivial). On the other hand, the left-hand-side
is also an inner product, this time on the
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although ...
. An application of the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, ...
for Hilbert spaces shows that there is a unique
solving (2) and therefore P1. This solution is a-priori only a member of
, but using
elliptic
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
regularity, will be smooth if
is.
Discretization
![Finite element method 1D illustration1](https://upload.wikimedia.org/wikipedia/commons/8/85/Finite_element_method_1D_illustration1.png)
P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:
:Find
such that
:
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
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
). However, for the finite element method we take
to be a space of piecewise polynomial functions.
For problem P1
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
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
.
For problem P2
We need
to be a set of functions of
. In the figure on the right, we have illustrated a
triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle me ...
of a 15 sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
al 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 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
. 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.
Choosing a basis
To complete the discretization, we must select a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of
. In the one-dimensional 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
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as ''th ...
. For the two-dimensional 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 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
hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size
''(h)'' and polynomial degree ' ...
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).
Small support of the basis
![Finite element solution](https://upload.wikimedia.org/wikipedia/commons/a/a9/Finite_element_solution.svg)
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 linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...
.)
In the one dimensional case, the
support of
is the interval