Galerkin method

The Galerkin method is a class of techniques for converting a continuous operator equation into a finite-dimensional system. An approximate solution is restricted to a finite-dimensional trial space, while the residual generated by that approximation is required to be orthogonal to a corresponding test space. The method provides a common mathematical structure for finite element methods, spectral methods, and several forms of boundary element analysis.

The defining orthogonality condition distinguishes the method from approaches based on pointwise enforcement. Rather than requiring the differential equation to hold at every location, a Galerkin formulation requires its integrated residual to vanish against each admissible test function. This replacement is particularly significant when the exact solution lacks the differentiability required by the original equation but remains meaningful as a weak solution.

Abstract formulation

Let (V) be a function space and consider the operator equation

[ \mathcal{A}u=f, ]

where (\mathcal{A}:V\rightarrow V') maps into an appropriate dual space. A finite-dimensional trial space (V_h\subset V) is introduced, with basis functions

[ {\phi_1,\phi_2,\ldots,\phi_n}. ]

The approximate solution has the representation

[ u_h=\sum_{j=1}^{n}c_j\phi_j. ]

Substitution into the governing equation produces the residual

[ R_h=f-\mathcal{A}u_h. ]

In the standard Galerkin method, the residual satisfies

[ \langle R_h,v_h\rangle=0 \qquad \text{for every }v_h\in V_h, ]

where the brackets denote either an inner product or a suitable duality pairing. Testing against every basis function yields the algebraic equations

[ \left\langle \mathcal{A}\sum_{j=1}^{n}c_j\phi_j,\phi_i\right\rangle

\langle f,\phi_i\rangle, \qquad i=1,\ldots,n. ]

For a linear operator, these relations form a matrix system

[ A\mathbf{c}=\mathbf{b}, ]

with entries

[ A_{ij}=\langle\mathcal{A}\phi_j,\phi_i\rangle, \qquad b_i=\langle f,\phi_i\rangle. ]

The resulting matrix inherits structural properties from the underlying operator and bilinear form. A symmetric coercive problem generally produces a symmetric positive-definite matrix when conforming trial spaces and exact integration are used.

Weak formulation

Many Galerkin schemes are formulated through a variational problem rather than through direct projection of a strong residual. Let (V) be a Hilbert space, let (a:V\times V\rightarrow\mathbb{R}) be a bilinear form, and let (F:V\rightarrow\mathbb{R}) be a continuous linear functional. The continuous weak problem is

[ a(u,v)=F(v) \qquad \text{for every }v\in V. ]

The discrete Galerkin problem seeks (u_h\in V_h) satisfying

[ a(u_h,v_h)=F(v_h) \qquad \text{for every }v_h\in V_h. ]

If (u) is the exact weak solution, subtraction of the continuous and discrete equations gives

[ a(u-u_h,v_h)=0 \qquad \text{for every }v_h\in V_h. ]

This relation is known as Galerkin orthogonality. It states that the approximation error is orthogonal to the discrete space in the geometry induced by the bilinear form. When (a) is an inner product, (u_h) is the corresponding orthogonal projection of (u) onto (V_h).

The weak formulation often transfers derivatives from the approximate solution to the test function through integration by parts. Boundary terms produced by this transformation encode natural boundary conditions, while essential boundary conditions are incorporated into the definition of the admissible function space.

Historical development

The method emerged from the interaction between weighted-residual techniques and variational approximation. In 1908, Walter Ritz formulated a finite-dimensional approach in which a stationary value of an energy functional determines the expansion coefficients of an approximate solution. The resulting Ritz method applies directly to problems possessing an appropriate variational principle.

Nikolai Bubnov subsequently used weighted-residual constructions in studies of elastic and structural systems. His work established the use of identical approximation and weighting functions in settings that were not expressed solely as the minimization of an energy functional.

In 1915, Boris Galerkin and You Watanabe presented the residual-orthogonality formulation in connection with boundary-value problems for elastic rods and plates. Their formulation expressed the governing equations through integrated residual conditions and reduced them to finite systems for the expansion coefficients. Later mathematical literature attached Galerkin’s surname to the general method, while the designation Bubnov–Galerkin method remained in use for formulations whose trial and test spaces coincide.

The subsequent development of functional analysis supplied an abstract setting for these constructions. Hilbert-space projection clarified the meaning of orthogonality, while the theory of weak derivatives extended the method to equations whose solutions need not possess classical derivatives of every order appearing in the strong formulation.

Relation to finite element discretization

In a finite element method, the Galerkin trial space is assembled from functions supported on a partition of the physical domain. Their local support generally produces a sparse system matrix because basis functions associated with distant regions do not interact in the bilinear form.

For a representative second-order elliptic equation,

[ -\nabla\cdot\bigl(k\nabla u\bigr)=f ]

on a domain (\Omega), multiplication by a test function (v) and integration by parts lead to the weak relation

[ \int_{\Omega}k\nabla u\cdot\nabla v,d\Omega

\int_{\Omega}fv,d\Omega + \int_{\partial\Omega_N}gv,dS, ]

where (g) denotes prescribed boundary flux on the Neumann portion of the boundary. A conforming finite element Galerkin approximation replaces the continuous space by a finite-dimensional subspace whose functions satisfy the essential boundary conditions.

The corresponding element matrices are assembled into a global matrix. This assembly reflects the decomposition of the domain rather than a decomposition of the differential operator itself. Numerical quadrature may alter the exact bilinear form, producing a related discrete problem whose consistency depends on the integration rule and the polynomial structure of the basis.

Approximation and convergence

For a continuous and coercive bilinear form, the Lax–Milgram theorem establishes existence and uniqueness of the weak solution. Under the same assumptions, Céa’s lemma bounds the Galerkin error by the best approximation error available in the discrete space:

[ |u-u_h|V \leq \frac{M}{\alpha} \inf{w_h\in V_h}|u-w_h|_V, ]

where (M) is the continuity constant and (\alpha) is the coercivity constant of the bilinear form. The estimate separates the stability of the variational problem from the approximation properties of the chosen space.

Convergence therefore follows when the discrete spaces approximate the continuous solution increasingly well and the stability constants remain controlled. The rate depends on the regularity of the exact solution, the degree of the basis functions, and the geometric quality of the domain approximation. Singularities in the solution can reduce the global rate even when the discrete formulation remains stable.

For noncoercive problems, Galerkin orthogonality alone does not ensure stable approximation. Mixed formulations and indefinite equations are instead governed by an inf–sup condition, which controls the interaction between trial and test spaces.

Petrov–Galerkin formulations

The Petrov–Galerkin method uses a trial space (V_h) and a distinct test space (W_h). Its discrete equations take the form

[ a(u_h,w_h)=F(w_h) \qquad \text{for every }w_h\in W_h. ]

The standard Galerkin method is recovered when (W_h=V_h). Distinct spaces alter the projection geometry and may change the stability properties of the discretization. This distinction is central for transport-dominated operators, whose standard symmetric weighting can produce oscillatory discrete solutions when the mesh does not resolve the relevant length scales.

Related stabilized formulations modify either the test functions or the weak equation. Such modifications retain a residual-based structure while replacing ordinary Galerkin orthogonality with an orthogonality relation defined by the altered test space or bilinear form.

Spectral interpretation

When the basis consists of globally supported orthogonal functions, the same projection principle produces a spectral Galerkin method. The approximation coefficients then represent global modes rather than locally supported finite element degrees of freedom. Smooth solutions can exhibit rapid coefficient decay, whereas limited regularity constrains the attainable convergence rate.

For a self-adjoint operator with a complete eigenfunction system, expansion in operator eigenfunctions can diagonalize the Galerkin matrix. More general bases produce coupled coefficient equations, although orthogonal polynomial systems often preserve substantial algebraic structure. The distinction between finite element and spectral realizations therefore concerns the construction of the discrete space rather than the defining Galerkin condition.