What substitution is made in Galerkin's Method?

What substitution is made in Galerkin's Method?

• A. wb = Na
• B. wb = Nb
• C. wb = Nb^2
• D. wb = NaNb

Answer: (B)

The key idea in Galerkin's method is to approximate the unknown field with a linear combination of basis (shape) functions and to enforce the residual to be orthogonal to the same subspace spanned by those basis functions. This means you use the same functions for both building the approximate solution and testing the residual. Concretely, each weighting (test) function is taken from the same set as the basis functions used to approximate the solution, so the weighting function for a given index matches the corresponding basis function: w with index b is set equal to N with index b. This choice ensures the residual is projected onto the same subspace that you’re using to approximate the solution, leading to a consistent, often symmetric system of equations and good convergence properties as you refine the basis. If you used a different test function (not from the same set), you’d be moving into a different method (like a Petrov–Galerkin approach), which changes stability and accuracy characteristics.

The key idea in Galerkin's method is to approximate the unknown field with a linear combination of basis (shape) functions and to enforce the residual to be orthogonal to the same subspace spanned by those basis functions. This means you use the same functions for both building the approximate solution and testing the residual. Concretely, each weighting (test) function is taken from the same set as the basis functions used to approximate the solution, so the weighting function for a given index matches the corresponding basis function: w with index b is set equal to N with index b.

This choice ensures the residual is projected onto the same subspace that you’re using to approximate the solution, leading to a consistent, often symmetric system of equations and good convergence properties as you refine the basis. If you used a different test function (not from the same set), you’d be moving into a different method (like a Petrov–Galerkin approach), which changes stability and accuracy characteristics.