Weak formulation

Weak formulation   (Redirected from Lax–Milgram theorem) Jump to navigation Jump to search 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 conditions are no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.

The Lax–Milgram theorem, named after Peter Lax and Arthur Milgram who proved it in 1954, provides weak formulations for certain systems on Hilbert spaces.

Contents 1 General concept 2 Example 1: linear system of equations 3 Example 2: Poisson's equation 4 The Lax–Milgram theorem 4.1 Application to example 1 4.2 Application to example 2 5 See also 6 References 7 External links General concept Let {displaystyle V} be a Banach space. To find the solution {displaystyle uin V} of the equation {displaystyle Au=f,} where {displaystyle Acolon Vto V'} and {displaystyle fin V'} , with {displaystyle V'} being the dual space of {displaystyle V} , is equivalent to finding {displaystyle uin V} such that for all {displaystyle vin V} , {displaystyle [Au](v)=f(v).} Here, {displaystyle v} is called a test vector or test function.

To bring this into the generic form of a weak formulation, find {displaystyle uin V} such that {displaystyle a(u,v)=f(v)quad forall vin V,} by defining the bilinear form {displaystyle a(u,v):=[Au](v).} Example 1: linear system of equations Now, let {displaystyle V=mathbb {R} ^{n}} and {displaystyle A:Vto V} be a linear mapping. Then, the weak formulation of the equation {displaystyle Au=f} involves finding {displaystyle uin V} such that for all {displaystyle vin V} the following equation holds: {displaystyle langle Au,vrangle =langle f,vrangle ,} where {displaystyle langle cdot ,cdot rangle } denotes an inner product.

Since {displaystyle A} is a linear mapping, it is sufficient to test with basis vectors, and we get {displaystyle langle Au,e_{i}rangle =langle f,e_{i}rangle ,quad i=1,ldots ,n.} Actually, expanding {displaystyle u=sum _{j=1}^{n}u_{j}e_{j}} , we obtain the matrix form of the equation {displaystyle mathbf {A} mathbf {u} =mathbf {f} ,} where {displaystyle a_{ij}=langle Ae_{j},e_{i}rangle } and {displaystyle f_{i}=langle f,e_{i}rangle } .

The bilinear form associated to this weak formulation is {displaystyle a(u,v)=mathbf {v} ^{T}mathbf {A} mathbf {u} .} Example 2: Poisson's equation To solve Poisson's equation {displaystyle -nabla ^{2}u=f,} on a domain {displaystyle Omega subset mathbb {R} ^{d}} with {displaystyle u=0} on its boundary, and to specify the solution space {displaystyle V} later, one can use the {displaystyle L^{2}} -scalar product {displaystyle langle u,vrangle =int _{Omega }uv,dx} to derive the weak formulation. Then, testing with differentiable functions {displaystyle v} yields {displaystyle -int _{Omega }(nabla ^{2}u)v,dx=int _{Omega }fv,dx.} The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that {displaystyle v=0} on {displaystyle partial Omega } : {displaystyle int _{Omega }nabla ucdot nabla v,dx=int _{Omega }fv,dx.} This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space {displaystyle V} must be zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space {displaystyle H_{0}^{1}(Omega )} of functions with weak derivatives in {displaystyle L^{2}(Omega )} and with zero boundary conditions, so {displaystyle V=H_{0}^{1}(Omega )} .

The generic form is obtained by assigning {displaystyle a(u,v)=int _{Omega }nabla ucdot nabla v,dx} and {displaystyle f(v)=int _{Omega }fv,dx.} The Lax–Milgram theorem This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form.

Let {displaystyle V} be a Hilbert space and {displaystyle a(cdot ,cdot )} a bilinear form on {displaystyle V} , which is bounded: {displaystyle |a(u,v)|leq C|u||v|,;} and coercive: {displaystyle a(u,u)geq c|u|^{2},.} Then, for any {displaystyle fin V'} , there is a unique solution {displaystyle uin V} to the equation {displaystyle a(u,v)=f(v)quad forall vin V} and it holds {displaystyle |u|leq {frac {1}{c}}|f|_{V'},.} Application to example 1 Here, application of the Lax–Milgram theorem is a stronger result than is needed.

Boundedness: all bilinear forms on {displaystyle mathbb {R} ^{n}} are bounded. In particular, we have {displaystyle |a(u,v)|leq |A|,|u|,|v|} Coercivity: this actually means that the real parts of the eigenvalues of {displaystyle A} are not smaller than {displaystyle c} . Since this implies in particular that no eigenvalue is zero, the system is solvable.

Additionally, this yields the estimate {displaystyle |u|leq {frac {1}{c}}|f|,} where {displaystyle c} is the minimal real part of an eigenvalue of {displaystyle A} .

Application to example 2 Here, choose {displaystyle V=H_{0}^{1}(Omega )} with the norm {displaystyle |v|_{V}:=|nabla v|,} where the norm on the right is the {displaystyle L^{2}} -norm on {displaystyle Omega } (this provides a true norm on {displaystyle V} by the Poincaré inequality). But, we see that {displaystyle |a(u,u)|=|nabla u|^{2}} and by the Cauchy–Schwarz inequality, {displaystyle |a(u,v)|leq |nabla u|,|nabla v|} .

Therefore, for any {displaystyle fin [H_{0}^{1}(Omega )]'} , there is a unique solution {displaystyle uin V} of Poisson's equation and we have the estimate {displaystyle |nabla u|leq |f|_{[H_{0}^{1}(Omega )]'}.} See also Babuška–Lax–Milgram theorem Lions–Lax–Milgram theorem References Lax, Peter D.; Milgram, Arthur N. (1954), "Parabolic equations", Contributions to the theory of partial differential equations, Annals of Mathematics Studies, vol. 33, Princeton, N. J.: Princeton University Press, pp. 167–190, doi:10.1515/9781400882182-010, MR 0067317, Zbl 0058.08703 External links MathWorld page on Lax–Milgram theorem Categories: Partial differential equationsNumerical differential equations

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