Weak Formulation and Galerkin's Weighted Residual Method The weak formulation and Galerkin's weighted residual method are two powerful techniques used in the...
The weak formulation and Galerkin's weighted residual method are two powerful techniques used in the Finite Element Method (FEM) to approximate solutions to differential equations. These methods involve representing the solution as a weighted sum of basis functions, and then minimizing the residual (the difference between the actual solution and the approximate solution) by adjusting the weights of these basis functions.
Weak Formulation:
Start by choosing a finite set of basis functions (e.g., linear polynomials, trigonometric functions, etc.).
Define the solution as a weighted sum of these basis functions, with the weights being the unknown parameters to be determined.
The weak formulation minimizes the residual by solving for the optimal weights that minimize the overall error.
This approach leads to an algebraic expression for the solution, which can be solved explicitly.
Galerkin's Weighted Residual Method:
Choose a sequence of weighted polynomials (e.g., polynomials with degree less than the original function).
Construct the solution as a weighted sum of these polynomials, with the weights being determined by the problem parameters.
Minimize the residual by adjusting the weights to minimize the error.
This approach leads to a system of weighted least-squares equations that can be solved iteratively.
Comparison:
The weak formulation is generally more efficient than Galerkin's method, especially for high-dimensional problems.
However, the weak formulation can be more difficult to solve due to the presence of multiple unknowns and the non-linearity of the basis functions.
Galerkin's method is more robust and can be applied to solve problems with complex boundary conditions.
Additional Notes:
Both methods rely on the Galerkin principle, which states that the solution can be expressed as a weighted sum of a finite set of functions.
They also share some similarities with the finite element method itself, which involves approximating the solution using a set of basis functions.
The weak formulation is a generalization of the Galerkin method that can be applied to more complex boundary conditions