Method for solving continuous operator problems (such as differential equations)
Differential equations
Scope
Fields
Natural sciences
Engineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations
Classification
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay
Solution
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters
People
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank
v
t
e
In mathematics, in the area of numerical analysis, Galerkin methods are named after the Soviet mathematician Boris Galerkin. They convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions.
Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used:
Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions.[1]
Bubnov–Galerkin method (after Ivan Bubnov) does not require the bilinear form to be symmetric and substitutes the energy minimization with orthogonality constraints determined by the same basis functions that are used to approximate the solution. In an operator formulation of the differential equation, Bubnov–Galerkin method can be viewed as applying an orthogonal projection to the operator.
Petrov–Galerkin method (after Georgii I. Petrov[2]) allows using basis functions for orthogonality constraints (called test basis functions) that are different from the basis functions used to approximate the solution. Petrov–Galerkin method can be viewed as an extension of Bubnov–Galerkin method, applying a projection that is not necessarily orthogonal in the operator formulation of the differential equation.
Examples of Galerkin methods are:
the Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method,[3][4]
the boundary element method for solving integral equations,
Krylov subspace methods.[5]
^A. Ern, J.L. Guermond, Theory and practice of finite elements, Springer, 2004, ISBN 0-387-20574-8
^"Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015
^S. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition, Springer, 2005, ISBN 0-387-95451-1
^P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978, ISBN 0-444-85028-7
^Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003, ISBN 0-89871-534-2
mathematics, in the area of numerical analysis, Galerkinmethods are named after the Soviet mathematician Boris Galerkin. They convert a continuous operator problem...
In applied mathematics, discontinuous Galerkinmethods (DG methods) form a class of numerical methods for solving differential equations. They combine...
approximation in this process, the finite element method is commonly introduced as a special case of Galerkinmethod. The process, in mathematical language, is...
Boris Grigoryevich Galerkin (Russian: Бори́с Григо́рьевич Галёркин, surname more accurately romanized as Galyorkin; 4 March [O.S. 20 February] 1871–12...
is the Hybrid-Collocation-Galerkinmethod (HCGM), which applies collocation at the interior Lobatto points and uses a Galerkin-like integral procedure at...
Some alternative formulations include the Rayleigh–Ritz method and the Ritz-Galerkinmethod. In quantum mechanics, a system of particles can be described...
accomplished either with collocation or a Galerkin or a Tau approach . For very small problems, the spectral method is unique in that solutions may be written...
Galerkin may refer to: Boris GalerkinGalerkinmethod, a method for discretisation of continuous problems, named after Boris Galerkin This disambiguation...
Petrov-GalerkinMethod: This method is similar to the Bubnov-Galerkin approach but differs in the choice of test functions. In the Petrov-Galerkinmethod, the...
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary...
the Galerkinmethod uses the test functions: w i = ∂ u ∂ a i {\displaystyle w_{i}={\frac {\partial u}{\partial a_{i}}}} The pseudospectral method which...
different methods need to be used to solve BVPs. For example, the shooting method (and its variants) or global methods like finite differences, Galerkin methods...
1990s a new class of meshfree methods emerged based on the Galerkinmethod. This first method called the diffuse element method (DEM), pioneered by Nayroles...
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives...
mainly applied to design stabilized finite element methods in which stability of the standard Galerkinmethod is not ensured both in terms of singular perturbation...
identity matrix. These equations are essentially a special case of a Galerkinmethod applied to the Hartree–Fock equation using a particular basis set....
form of the method in which the integrals over the source and field patches are the same is called "Galerkin'smethod". Galerkin'smethod is the obvious...
(integrations by parts) and is applicable to non-nested geometries. When the Galerkinmethod is applied and the same zeroth-order basis functions (with a constant...
gyroscope, a heading indicator use in aircraft Discontinuous Galerkinmethod, a numerical method Distributed generation of energy Cebgo, formerly South East...
; Doblaré, M. (2005). "A natural neighbour Galerkinmethod with quadtree structure". Int. J. Numer. Methods Eng. 63 (6): 789–812. Bibcode:2005IJNME..63...
element methodsGalerkinmethod — a finite element method in which the residual is orthogonal to the finite element space Discontinuous Galerkinmethod — a...
first to catalogue all 230 space groups of crystals Boris Galerkin, developed the Galerkinmethod in numerical analysis Israel Gelfand, major contributor...
Global Information