Generalized Finite Element Methods Approximate solution techniques Rayleigh

Generalized Finite Element Methods Approximate solution techniques: Rayleigh. Ritz and Galerkin Methods Suvranu De

Last class Strong formulation (BVP) Minimization statement Weak formulation (VBVP)

This class Approximate solution techniques: Rayleigh Ritz Method Equivalence Galerkin Method Other techniques

The Poisson equation The strong formulation (BVP) Domain: for given f(x) Dirichlet problem

The Dirichlet problem Find where X= and Minimization Principle Statement. . .

The Dirichlet problem Let a SPD bilinear form and a linear form Weak formulation (VBVP) Restatement. . .

The Dirichlet problem Weak formulation (VBVP) Minimization Principle: Weak Statement (VBVP): Find Restatement. . .

The Dirichlet problem in 2 D Strong form Find u(x, y) such that W where and W is a domain in R 2 with boundary G G

The Dirichlet problem in 2 D Weak formulation Restatement

Approximation Xh X Basis Define a finite dimensional subspace Xh of X spanned by linearly independent functions “basis functions” i. e. , any function wh Xh may be written as dim(ension) (Xh) = N

Xh X Approximation These functions Basis can be Lagrange polynomials Least squares Moving least squares functions PU functions Wavelets …. .

The Rayleigh-Ritz Method Minimization Principle General approach Start with the minimization principle and pose it in the subspace Xh (APPROXIMATION)

The Rayleigh-Ritz Method Minimization Principle Approximate solution in the subspace Xh is General approach

The Rayleigh-Ritz Method Minimization Principle Geometric interpretation

The Rayleigh-Ritz Method Minimization Principle J|Xh. . . Claim: For problems of type minimization of the functional in the subspace Xh is equivalent to the following problem Find such that

The Rayleigh-Ritz Method Using the property of bilinearity Minimization Principle J|Xh. . .

The Rayleigh-Ritz Method Using the property of linearity Hence Minimization Principle J|Xh. . .

The Rayleigh-Ritz Method For our Dirichlet problem Notice that Ah is SPD Minimization Principle J|Xh. . .

The Rayleigh-Ritz Method Minimization Principle J|Xh. . .

The Rayleigh-Ritz Method Minimization Principle J|Xh. . .

The Rayleigh-Ritz Method Minimization Principle J|Xh. . . Since Ah is symmetric Similarly

The Rayleigh-Ritz Method Minimization Principle J|Xh. . . • Different bases will generate different Ah - with different bandedness, sparsity and conditioning - and hence different solution. • Since Ah is SPD for the Dirichlet problem, the solution always exists and is unique.

The Galerkin Method VBVP • Takes the weak form (VBVP) as the starting point. • Very general. Works for problems that does not have a minimization principle. • For problems that do have a minimization principle, the Galerkin method and the Rayleigh-Ritz methods produce exactly the same set of discrete equations.

The Galerkin Method VBVP Approximation Pose the weak form: Find such that in the subspace Xh (APPROXIMATION) Find such that

The Galerkin Method VBVP Using the property of bilinearity and linearity as before Using symmetricity of Ah

The Galerkin Method VBVP Discrete equations The discrete problem Find such that Choose First row

The Galerkin Method VBVP Discrete equations The discrete problem Find such that Choose Second row and so on. .

The Galerkin Method VBVP Discretized set of equations The discrete problem Find such that is equivalent to Find such that

The Galerkin Method VBVP Example Numerical solution Analytical solution

Summary • The Rayleigh-Ritz Method takes the Minimization principle as its starting point • The Galerkin Method takes the Weak form as its starting point • The Galerkin Method is much more general than the Rayleigh-Ritz Method. • For problems that permit a minimization principle, both the methods give rise to the same set of equations in the same finite dimensional subspace.