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

A direct solution of the strong form is difficult. It needs the field to be smooth enough for the second derivative to exist everywhere. The finite-element method works instead with a weak form. The weak form 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:

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

Integrate the left-hand side by parts (Green's identity). This 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, \(u = g\), apply directly to the solution. Choose the test functions to vanish on \(\Gamma_D\), so the boundary integral drops there. mufem enforces these later as constraints.
Natural (Neumann) conditions, \(\kappa\,\nabla u \cdot \mathbf{n} = q\), substitute into the boundary integral. It becomes a known term \(\int_{\Gamma_N} q\, v \, \mathrm{d}\Gamma\) on the right-hand side. The space needs no special treatment, 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 so to \(A\)), while the \(q\) part adds to the right-hand side. They model impedance-type boundaries, such as convective heat transfer. There \(\alpha\) 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 \(\mathbf{b}\). This gives \(A\,\mathbf{x} = \mathbf{b}\).