Weak Form¶

Strong form¶

A physical model is stated as a partial differential equation that holds at every point of the domain, its strong form. A representative example is steady diffusion, which governs thermal conduction, electrostatics, and many other scalar problems:

\[ -\nabla \cdot (\kappa\, \nabla u) = f \quad \text{in } \Omega , \]

with boundary conditions

\[ u = g \quad \text{on } \Gamma_D , \qquad \kappa\, \nabla u \cdot \mathbf{n} = q \quad \text{on } \Gamma_N . \]

Symbol	Meaning
\(u\)	Unknown field (temperature, electric potential, ...)
\(\kappa\)	Material coefficient (conductivity, permittivity, ...)
\(f\)	Volumetric source
\(\Omega\)	Computational domain, with boundary \(\Gamma = \Gamma_D \cup \Gamma_N\)
\(g\)	Prescribed value on the Dirichlet boundary \(\Gamma_D\)
\(q\)	Prescribed flux on the Neumann boundary \(\Gamma_N\)
\(\mathbf{n}\)	Outward unit normal

Solving the strong form directly is difficult: it requires the field to be smooth enough for the second derivative to exist everywhere. The finite-element method works instead with a weak form, which asks the equation to hold only in an averaged (integral) sense and needs one fewer derivative.

Weak form¶

Multiply the strong form by a test function \(v\) and integrate over the domain:

\[ -\int_\Omega \nabla \cdot (\kappa\, \nabla u)\, v \; \mathrm{d}\Omega = \int_\Omega f\, v \; \mathrm{d}\Omega . \]

Integrating the left-hand side by parts (Green's identity) moves one derivative from \(u\) onto \(v\) and produces a boundary term:

\[ \int_\Omega \kappa\, \nabla u \cdot \nabla v \; \mathrm{d}\Omega - \int_\Gamma (\kappa\, \nabla u \cdot \mathbf{n})\, v \; \mathrm{d}\Gamma = \int_\Omega f\, v \; \mathrm{d}\Omega . \]

Boundary conditions enter the weak form in different ways:

Essential (Dirichlet) conditions, \(u = g\), are imposed directly on the solution. The test functions are chosen to vanish on \(\Gamma_D\), so the boundary integral drops there. These are enforced later as constraints.
Natural (Neumann) conditions, \(\kappa\,\nabla u \cdot \mathbf{n} = q\), are simply substituted into the boundary integral, which becomes a known term \(\int_{\Gamma_N} q\, v \, \mathrm{d}\Gamma\) on the right-hand side. No special treatment of the space is needed, which is why they are called natural.
Robin (mixed) conditions, \(\kappa\,\nabla u \cdot \mathbf{n} = q - \alpha\, u\), also enter through the boundary integral, but because the flux depends on \(u\) itself they split into two parts. The \(\alpha\, u\) part adds a surface term \(\int_{\Gamma_R} \alpha\, u\, v \, \mathrm{d}\Gamma\) to the bilinear form (and hence to \(A\)), while the \(q\) part adds to the right-hand side. They model impedance-type boundaries, such as convective heat transfer, where \(\alpha\) is the heat-transfer coefficient and \(q = \alpha\, u_\infty\) carries the ambient temperature.

The result is the weak form: find \(u\) (with \(u = g\) on \(\Gamma_D\)) such that

\[ \underbrace{\int_\Omega \kappa\, \nabla u \cdot \nabla v \; \mathrm{d}\Omega}_{a(u,\,v)} = \underbrace{\int_\Omega f\, v \; \mathrm{d}\Omega + \int_{\Gamma_N} q\, v \; \mathrm{d}\Gamma}_{\ell(v)} \qquad \text{for all admissible } v . \]

Bilinear and linear forms¶

The weak form separates into two pieces:

the bilinear form \(a(u, v)\), linear in each of \(u\) and \(v\), which carries the operator and the material properties;
the linear form \(\ell(v)\), which carries the sources and the natural boundary data.

This split is exactly the structure of the discrete system. After discretization, the bilinear form becomes the system matrix \(A\) and the linear form becomes the right-hand side \(\mathbf{b}\), giving \(A\,\mathbf{x} = \mathbf{b}\).