Series: Finite Element Method › Continuous

Weak Formulation of the Poisson Problem

The strong form of the Poisson problem — find uu satisfying −Δu=f-\Delta u = f pointwise in Ω\Omega together with boundary conditions on ∂Ω\partial\Omega — requires uu to be twice continuously differentiable. This regularity is often unavailable for domains or data that arise in practice, and it is also inconvenient for the finite element method. The weak formulation (also called the variational formulation) relaxes this requirement and provides the correct mathematical framework for approximation.

The idea is to multiply the equation −Δu=f-\Delta u = f by a smooth test function vv and integrate over Ω\Omega:

−∫Ω(Δu) v dx=∫Ωf v dx.-\int_\Omega (\Delta u)\, v \, \mathrm{d}x = \int_\Omega f\, v \, \mathrm{d}x.

The left-hand side involves second derivatives of uu, which is precisely what we want to avoid. Applying Green's first identity (integration by parts in multiple dimensions),

−∫Ω(Δu) v dx=∫Ω∇u⋅∇v dx−∫∂Ω∂u∂n v ds,-\int_\Omega (\Delta u)\, v \, \mathrm{d}x = \int_\Omega \nabla u \cdot \nabla v \, \mathrm{d}x - \int_{\partial\Omega} \frac{\partial u}{\partial n}\, v \, \mathrm{d}s,

where n\mathbf{n} is the unit outward normal and ∂u/∂n=∇u⋅n\partial u / \partial n = \nabla u \cdot \mathbf{n} is the outward normal derivative. Combining with the right-hand side,

∫Ω∇u⋅∇v dx−∫∂Ω∂u∂n v ds=∫Ωf v dx.\int_\Omega \nabla u \cdot \nabla v \, \mathrm{d}x - \int_{\partial\Omega} \frac{\partial u}{\partial n}\, v \, \mathrm{d}s = \int_\Omega f\, v \, \mathrm{d}x.

Now the boundary conditions come into play. We choose the test function vv to vanish on the Dirichlet boundary: v=0v = 0 on ΓD\Gamma_D. This makes the Dirichlet part of the boundary integral disappear. On the Neumann boundary ΓN\Gamma_N the flux is prescribed as ∂u/∂n=gN\partial u / \partial n = g_N, so that part of the boundary integral is known. Moving it to the right-hand side gives

∫Ω∇u⋅∇v dx=∫Ωf v dx+∫ΓNgN v ds.\int_\Omega \nabla u \cdot \nabla v \, \mathrm{d}x = \int_\Omega f\, v \, \mathrm{d}x + \int_{\Gamma_N} g_N\, v \, \mathrm{d}s.

This is the weak form of the Poisson problem. Two things are worth noting. First, the equation now involves only first derivatives of both uu and vv, which considerably widens the class of admissible functions. Second, the Neumann condition has entered naturally: it was not imposed explicitly but appeared as a boundary integral from the integration-by-parts step. For this reason Neumann conditions are often called natural boundary conditions in the finite element literature.

To state the weak problem precisely we need to specify the function spaces. The natural space is the Sobolev space H1(Ω)H^1(\Omega), consisting of all square-integrable functions whose first-order partial derivatives are also square-integrable. We define the trial space

V={v∈H1(Ω):v=gD on ΓD}V = \{ v \in H^1(\Omega) : v = g_D \text{ on } \Gamma_D \}

and the test space

V0={v∈H1(Ω):v=0 on ΓD}.V_0 = \{ v \in H^1(\Omega) : v = 0 \text{ on } \Gamma_D \}.

The weak formulation then reads: find u∈Vu \in V such that

∫Ω∇u⋅∇v dx=∫Ωf v dx+∫ΓNgN v dsfor all v∈V0.\int_\Omega \nabla u \cdot \nabla v \, \mathrm{d}x = \int_\Omega f\, v \, \mathrm{d}x + \int_{\Gamma_N} g_N\, v \, \mathrm{d}s \quad \text{for all } v \in V_0.

The boundary values gDg_D are said to be imposed essentially (they restrict the space VV), while the Neumann data gNg_N appear in the right-hand side and are imposed naturally. When ΓD=∂Ω\Gamma_D = \partial\Omega (pure Dirichlet problem) and gD=0g_D = 0, the two spaces coincide: V=V0=H01(Ω)V = V_0 = H^1_0(\Omega).

Well-posedness of this problem — existence and uniqueness of u∈Vu \in V and continuous dependence on ff and gNg_N — follows from the Lax–Milgram theorem, a fundamental result in functional analysis. The left-hand side is a bounded, coercive bilinear form on H1(Ω)H^1(\Omega) and the right-hand side is a bounded linear functional, so the theorem applies. The coercivity here ultimately rests on the Poincaré inequality, which guarantees that ∥∇v∥L2(Ω)\|\nabla v\|_{L^2(\Omega)} controls ∥v∥H1(Ω)\|v\|_{H^1(\Omega)} for functions vanishing on ΓD\Gamma_D. A concise account of these results can be found in Evans, Partial Differential Equations (AMS, 2010), Chapter 6.

The weak formulation is the starting point for the finite element method. Instead of seeking uu in the infinite-dimensional space VV, we will restrict attention to a finite-dimensional subspace — spanned by carefully chosen basis functions defined on a mesh — and solve the resulting linear system. This is the subject of the next posts.