6.2 Weak Formulation and Variational Principles

Weak formulation and variational principles are key concepts in finite element methods. They transform differential equations into integral forms, making them easier to solve numerically. This approach relaxes continuity requirements and allows for a wider class of solutions.

The weak form uses test functions to enforce equations in a weighted integral sense. It reduces differentiation order, simplifies boundary conditions, and forms the basis for various numerical methods. Understanding these principles is crucial for applying finite element methods effectively.

Weak Form of Differential Equations

Derivation using Variational Principles

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• Multiply the differential equation by a test function and integrate over the domain
• Apply integration by parts to the highest order derivative term
  • Reduces the order of differentiation
  • Introduces boundary terms
• Weak form relaxes continuity requirements on the solution compared to the strong form
  • Allows for a wider class of solutions (piecewise continuous functions)
• Forms the foundation for various numerical methods (finite element method)

Properties and Advantages

• Weak form is an integral formulation of the differential equation
  • Enforces the equation in a weighted integral sense rather than pointwise
• Reduces the order of differentiation required for the solution
  • Enables the use of lower-order approximation spaces (piecewise linear functions)
• Incorporates natural boundary conditions directly into the formulation
  • Simplifies the treatment of boundary conditions in numerical methods

Test Functions in Weak Formulation

Definition and Role

• Test functions, also known as weight functions, are arbitrary functions used in the derivation of the weak form
• Belong to a suitable function space that satisfies certain continuity and boundary conditions
  • Typically required to be square-integrable and have square-integrable derivatives up to a certain order
• Used to enforce the differential equation in a weighted integral sense
  • Multiplied by the differential equation and integrated over the domain

Function Spaces and Properties

• Choice of test function space determines the properties of the weak form and the resulting numerical method
• Common test function spaces include:
  • $L^2(\Omega)$: square-integrable functions over the domain $\Omega$
  • $H^1(\Omega)$: functions in $L^2(\Omega)$ with square-integrable first derivatives
  • $H^1_0(\Omega)$: functions in $H^1(\Omega)$ that vanish on the boundary $\partial\Omega$
• Test functions are typically chosen to have compact support
  • Non-zero only on a small subset of the domain (finite element basis functions)
  • Enables local enforcement of the differential equation

Galerkin Method for Discretization

Approximation of Solution

• Galerkin method approximates the solution as a linear combination of basis functions from a finite-dimensional subspace
  • $u_h(x) = \sum_{i=1}^N c_i \phi_i(x)$, where $\phi_i(x)$ are the basis functions and $c_i$ are the coefficients
• Basis functions are chosen to satisfy the boundary conditions and have desired properties
  • Piecewise polynomials (linear, quadratic, cubic) over a mesh
  • Smooth functions with compact support (B-splines, NURBS)

Discrete System of Equations

• Weak form is discretized by substituting the approximate solution and test functions into the weak formulation
  • $\int_\Omega v_h (Lu_h - f) dx = 0$, where $L$ is the differential operator and $f$ is the source term
• Resulting discrete system of equations is obtained by requiring the weak form to hold for all test functions in the chosen subspace
  • Leads to a system of linear equations $Ac = b$, where $A$ is the stiffness matrix, $c$ is the vector of coefficients, and $b$ is the load vector
• Coefficients of the basis functions in the approximate solution are determined by solving the discrete system of equations
  • Can be solved using direct methods (Gaussian elimination) or iterative methods (conjugate gradient)

Weak Form vs Energy Minimization

Energy Functionals in Physical Problems

• Many physical problems can be formulated as the minimization of an energy functional
  • Potential energy in elasticity: $\Pi(u) = \frac{1}{2}\int_\Omega \sigma(u) : \varepsilon(u) dx - \int_\Omega f \cdot u dx - \int_{\partial\Omega} g \cdot u ds$
  • Total energy in heat transfer: $E(u) = \frac{1}{2}\int_\Omega \nabla u \cdot \kappa \nabla u dx - \int_\Omega fu dx - \int_{\partial\Omega} gu ds$
• Minimizing the energy functional leads to the equilibrium state or steady-state solution of the physical system
  • Principle of minimum potential energy in elasticity
  • Principle of minimum total potential energy in heat transfer

Euler-Lagrange Equation and Weak Form

• Euler-Lagrange equation, derived from variational principles, provides the necessary condition for a function to minimize an energy functional
  • $\frac{\partial L}{\partial u} - \frac{d}{dx}\left(\frac{\partial L}{\partial u'}\right) = 0$, where $L(u, u')$ is the Lagrangian density
• Weak form of a differential equation often corresponds to the Euler-Lagrange equation for a specific energy functional
  • Weak form of the Poisson equation: $\int_\Omega \nabla v \cdot \nabla u dx = \int_\Omega vf dx$
  • Euler-Lagrange equation for the total potential energy functional: $-\nabla \cdot (\kappa \nabla u) = f$ in $\Omega$, $\kappa \nabla u \cdot n = g$ on $\partial\Omega$
• Minimizing the energy functional is equivalent to finding the solution of the weak form
  • Galerkin method can be interpreted as a Ritz method for minimizing the energy functional

Physical Interpretation and Insights

• Connection between weak form and energy minimization allows for the interpretation of the weak form as a minimization problem
  • Solution of the weak form corresponds to the minimum of the associated energy functional
• Provides insights into the physical meaning of the solution
  • Displacement field that minimizes the potential energy in elasticity
  • Temperature distribution that minimizes the total energy in heat transfer
• Enables the use of optimization techniques for solving the weak form
  • Gradient descent methods
  • Newton's method for minimization