Strong and Weak form

Editor: Weipeng Xu & Copilot

Last modified: 24/01/2026

In the context of numerical methods for solving partial differential equations (PDEs), the strong form and weak form refer to two different ways of expressing the governing equations that describe the behavior of materials under various conditions. For the strong form, the governing equations are expressed in terms of differential equations that must be satisfied at every point within the domain of interest. The weak form, on the other hand, is derived from the strong form by multiplying the governing equations by a set of test functions and integrating over the domain. This process leads to an integral equation that must be satisfied in an average sense over the domain, rather than at every point. Besides, the requirements for the continuity of the solution are relaxed in the weak form, allowing for solutions that may not be differentiable everywhere.

Formulations

For the static solid mechanics problem, the strong form can be expressed as: $$ \begin{align*} \nabla \cdot \boldsymbol{\sigma} + \boldsymbol{b} = \boldsymbol{0} \quad &\text{in } \Omega \\ \boldsymbol{u} = \boldsymbol{g} \quad &\text{on } \Gamma_D \\ \boldsymbol{\sigma} \cdot \boldsymbol{n} = \boldsymbol{t} \quad &\text{on } \Gamma_N \end{align*} $$ where $\Omega$ is the problem domain of interest. $\boldsymbol{\sigma}$ is the Cauchy stress tensor, $\boldsymbol{b}$ is the body force per unit volume, $\boldsymbol{u}$ is the displacement vector, $\boldsymbol{g}$ is the prescribed displacement on the Dirichlet boundary $\Gamma_D$, $\boldsymbol{t}$ is the prescribed traction on the Neumann boundary $\Gamma_N$, and $\boldsymbol{n}$ is the outward normal vector to the boundary.

The solution to the strong form must satisfy the differential equations and boundary conditions pointwise within the domain and on its boundaries, which poses strict requirements on the smoothness and continuity of the solution. On the other hand, the weak form relaxes these requirements by seeking solutions $\boldsymbol{u}\in U$ that satisfy the integral form of the governing equations, where $U=\{\boldsymbol{u} | \boldsymbol{u} \in H^1(\Omega), \boldsymbol{u}=\boldsymbol{g} \text{ on } \Gamma_D\}$ is the space of admissible functions that meet the Dirichlet boundary conditions.

The weak form of the same problem can be derived by multiplying the strong form by a test function $\boldsymbol{v}\in V$ and integrating over the domain $\Omega$: $$ \int_{\Omega} \boldsymbol{v} \cdot (\nabla \cdot \boldsymbol{\sigma} + \boldsymbol{b}) \, d\Omega = 0 $$ where $V=\{\boldsymbol{v} | \boldsymbol{v} \in H^1(\Omega), \boldsymbol{v}=\boldsymbol{0} \text{ on } \Gamma_D\}$ is the space of admissible test functions that vanish on the Dirichlet boundary. We preform integration by parts on the first term: $$ \int_{\Omega} \nabla \boldsymbol{v} : \boldsymbol{\sigma} \, d\Omega - \int_{\Omega}\nabla \cdot (\boldsymbol{v}\cdot\boldsymbol{\sigma}) \, d\Omega + \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega = 0 $$ Then we apply the divergence theorem: $$ \int_{\Omega} \nabla \boldsymbol{v} : \boldsymbol{\sigma} \, d\Omega - \int_{\Gamma} \boldsymbol{v} \cdot (\boldsymbol{\sigma} \cdot \boldsymbol{n}) \, d\Gamma + \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega = 0 $$ Substituting the boundary conditions, we obtain the weak form:

Find $\boldsymbol{u}\in U$ such that for all $\boldsymbol{v}\in V$, $$ \int_{\Omega} \nabla \boldsymbol{v} : \boldsymbol{\sigma} \, d\Omega = \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} \boldsymbol{v} \cdot \boldsymbol{t} \, d\Gamma $$

Alternative perspectives

Above we derived the weak form through integration by parts. Alternatively, we can also derive the weak form using other approaches, such as the variational formulation and the principle of virtual work. Here we briefly outline these two methods.

Variational analysis

The variational method involves formulating the problem in terms of minimizing an energy functional. The mechanical system is assumed to be conservative and the potential energy of the system is defined, such as the linear elasticity and hyperelasticity problems. The energy functional for the static solid mechanics problem can be expressed as: $$ \Pi(\boldsymbol{u}) = U(\boldsymbol{u}) - W_{ext}(\boldsymbol{u}) $$ where $U=\int_{\Omega} W(\boldsymbol{\varepsilon}) \, d\Omega$ is the strain energy stored in the body, with $W(\boldsymbol{\varepsilon})$ being the strain energy density function, and $W_{ext}=\int_{\Omega} \boldsymbol{u} \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} \boldsymbol{u} \cdot \boldsymbol{t} \, d\Gamma$ is the work done by external forces. The principle of minimum potential energy states that the actual displacement field $\boldsymbol{u}$ minimizes the potential energy functional $\Pi(\boldsymbol{u})$: $$ \delta \Pi(\boldsymbol{u}; \boldsymbol{v}) = \delta U(\boldsymbol{u}; \boldsymbol{v}) - \delta W_{ext}(\boldsymbol{u}; \boldsymbol{v}) = 0, \quad \forall \boldsymbol{v} \in V $$ where $\boldsymbol{v}$ is an admissible variation of the displacement field that vanishes on the Dirichlet boundary $\Gamma_D$. Thus, we take the first variation of $\Pi$ (Gateaux derivative) and set it to zero: $$ \delta \Pi(\boldsymbol{u}; \boldsymbol{v}) = \lim_{\epsilon \to 0} \frac{\Pi(\boldsymbol{u} + \epsilon \boldsymbol{v}) - \Pi(\boldsymbol{u})}{\epsilon} = 0 $$ The variation of $U$ is: $$ \delta U(\boldsymbol{u}; \boldsymbol{v}) = \left.\frac{d}{d\epsilon}\right|_{\epsilon=0} \int_{\Omega} W(\boldsymbol{\varepsilon}(\boldsymbol{u} + \epsilon \boldsymbol{v})) \, d\Omega = \int_{\Omega} \frac{\partial W}{\partial \boldsymbol{\varepsilon}} : \left.\frac{\partial \boldsymbol{\varepsilon}}{\partial \epsilon}\right|_{\epsilon=0} \, d\Omega = \int_{\Omega} \boldsymbol{\sigma} : \left.\frac{\partial \boldsymbol{\varepsilon}}{\partial \epsilon}\right|_{\epsilon=0} \, d\Omega $$ where $$ \left.\frac{\partial \boldsymbol{\varepsilon}}{\partial \epsilon}\right|_{\epsilon=0} = \left.\frac{1}{2}(\nabla (\boldsymbol{u}+\epsilon \boldsymbol{v}) + (\nabla (\boldsymbol{u}+\epsilon \boldsymbol{v}))^T)\right|_{\epsilon=0} = \frac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T) $$ Since $\boldsymbol{\sigma}$ is symmetric, we have $$ \boldsymbol{\sigma} : \nabla \boldsymbol{v} = \boldsymbol{\sigma} : \frac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T) + \boldsymbol{\sigma} : \frac{1}{2}(\nabla \boldsymbol{v} - (\nabla \boldsymbol{v})^T) = \boldsymbol{\sigma} : \frac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T) $$ Thus, we can write the variation of $U$ as: $$ \delta U(\boldsymbol{u}; \boldsymbol{v}) = \int_{\Omega} \boldsymbol{\sigma} : \nabla \boldsymbol{v} \,d\Omega $$

The variation of $W_{ext}$ is: $$ \delta W_{ext}(\boldsymbol{u}; \boldsymbol{v}) = \left.\frac{d}{d\epsilon}\right|_{\epsilon=0} \left( \int_{\Omega} (\boldsymbol{u} + \epsilon \boldsymbol{v}) \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} (\boldsymbol{u} + \epsilon \boldsymbol{v}) \cdot \boldsymbol{t} \, d\Gamma \right) = \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} \boldsymbol{v} \cdot \boldsymbol{t} \, d\Gamma $$

Finally, substituting the variations of $U$ and $W_{ext}$ into the expression for $\delta \Pi$ and setting it to zero, we obtain the weak form: $$ \int_{\Omega} \boldsymbol{\sigma} : \nabla \boldsymbol{v} \, d\Omega = \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} \boldsymbol{v} \cdot \boldsymbol{t} \, d\Gamma, \quad \forall \boldsymbol{v} \in V $$

Principle of virtual work

The principle of virtual work states that for a system in equilibrium, the total virtual work done by internal forces must equal the total virtual work done by external forces for any virtual displacement field that is kinematically admissible (i.e., satisfies the boundary conditions). $$ \delta W_{int} = \delta W_{ext} \, \quad \forall \boldsymbol{v} \in V $$ where $\boldsymbol{v}$ is a virtual displacement field that vanishes on the Dirichlet boundary $\Gamma_D$.

The virtual work done by internal forces can be expressed as: $$ \delta W_{int} = \int_{\Omega} \boldsymbol{\sigma} : \delta \boldsymbol{\varepsilon} \, d\Omega $$ where $\delta \boldsymbol{\varepsilon}$ is the virtual strain corresponding to the virtual displacement field $\boldsymbol{v}$. Again, since $\boldsymbol{\sigma}$ is symmetric, we have $$ \boldsymbol{\sigma} : \nabla \boldsymbol{v} = \boldsymbol{\sigma} : \frac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T) + \boldsymbol{\sigma} : \frac{1}{2}(\nabla \boldsymbol{v} - (\nabla \boldsymbol{v})^T) = \boldsymbol{\sigma} : \frac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T) = \delta \boldsymbol{\varepsilon} $$ Thus, we can write the virtual work done by internal forces as: $$ \delta W_{int} = \int_{\Omega} \boldsymbol{\sigma} : \nabla \boldsymbol{v} \, d\Omega $$

The virtual work done by external forces is given by: $$ \delta W_{ext} = \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} \boldsymbol{v} \cdot \boldsymbol{t} \, d\Gamma $$

Substituting the expressions for $\delta W_{int}$ and $\delta W_{ext}$ into the principle of virtual work, we obtain: $$ \int_{\Omega} \boldsymbol{\sigma} : \nabla \boldsymbol{v} \, d\Omega = \int_{\Omega} \boldsymbol{v} \cdot \boldsymbol{b} \, d\Omega + \int_{\Gamma_N} \boldsymbol{v} \cdot \boldsymbol{t} \, d\Gamma, \quad \forall \boldsymbol{v} \in V $$

Further discussion

Relationship between other methods

Function spaces

Integration by parts

Principle of minimum potential energy

Among all kinematically admissible displacement fields that satisfy the displacement boundary conditions and compatibility conditions, the actual displacement field renders the total potential energy of the system stationary, and this stationary value is a minimum.