Series: Finite Element Method › Continuous

Weak Formulation of Elliptic PDEs

The weak formulation of the Poisson problem extends naturally to the general elliptic operator

$$Lu = -\nabla \cdot (A \nabla u) + \mathbf{b} \cdot \nabla u + c\, u.$$

The trial set $V$ and the test space $V_0$ are the same as before, consisting of the functions in $H^1(\Omega)$ that attain the prescribed values $g_D$ on $\Gamma_D$ and that vanish there, respectively. The derivation also follows the same steps: multiply $Lu = f$ by a test function $v \in V_0$ and integrate over $\Omega$,

$$\int_\Omega \bigl(-\nabla \cdot (A \nabla u) + \mathbf{b} \cdot \nabla u + c\, u\bigr)\, v \, \mathrm{d}x = \int_\Omega f\, v \, \mathrm{d}x.$$

Applying Green's first identity to the divergence term,

$$-\int_\Omega \nabla \cdot (A \nabla u)\, v \, \mathrm{d}x = \int_\Omega (A \nabla u) \cdot \nabla v \, \mathrm{d}x - \int_{\partial\Omega} (A \nabla u) \cdot \mathbf{n}\, v \, \mathrm{d}s.$$

The boundary integral vanishes on $\Gamma_D$ because $v = 0$ there, and on $\Gamma_N$ the Neumann condition prescribes the conormal derivative, $(A \nabla u) \cdot \mathbf{n} = g_N$, which takes over the role played by the normal derivative in the Poisson problem. Moving this known quantity to the right-hand side and collecting all terms gives

$$\int_\Omega \bigl[(A \nabla u) \cdot \nabla v + (\mathbf{b} \cdot \nabla u)\, v + c\, u\, v\bigr] \mathrm{d}x = \int_\Omega f\, v \, \mathrm{d}x + \int_{\Gamma_N} g_N\, v \, \mathrm{d}s.$$

It is conventional to name the two sides separately. Define the bilinear form $a : H^1(\Omega) \times V_0 \to \mathbb{R}$ by

$$a(u, v) = \int_\Omega \bigl[(A \nabla u) \cdot \nabla v + (\mathbf{b} \cdot \nabla u)\, v + c\, u\, v\bigr] \mathrm{d}x,$$

and the linear functional $\ell : V_0 \to \mathbb{R}$ by

$$\ell(v) = \int_\Omega f\, v \, \mathrm{d}x + \int_{\Gamma_N} g_N\, v \, \mathrm{d}s.$$

The weak problem then takes the compact abstract form: find $u \in V$ such that

$$a(u, v) = \ell(v) \quad \text{for all } v \in V_0.$$

The Poisson problem is recovered by setting $A = I$, $\mathbf{b} = 0$, $c = 0$, which gives $a(u, v) = \int_\Omega \nabla u \cdot \nabla v \, \mathrm{d}x$. The abstract notation $a(u,v) = \ell(v)$ is standard throughout the finite element literature and applies equally to far more general problems.

Well-posedness again follows from the Lax–Milgram theorem, which asks for two bounds on $a$ beyond bilinearity. It is continuous if there is a constant $M > 0$ with

$$|a(u, v)| \leq M\, \|u\|_V \|v\|_V \quad \text{for all } u, v \in V_0,$$

and coercive if there is a constant $\alpha > 0$ with

$$a(v, v) \geq \alpha\, \|v\|_V^2 \quad \text{for all } v \in V_0,$$

where $\|\cdot\|_V$ is the $H^1(\Omega)$ norm. Given these, together with $\ell$ bounded on $V_0$, Lax–Milgram guarantees a unique $u \in V$ solving $a(u,v) = \ell(v)$ for all $v \in V_0$, depending continuously on the data. These two constants reappear later in the series, when the quality of the finite element approximation is measured against them.

For the operator $L$ above, continuity and coercivity follow from the conditions listed for the Poisson problem together with two requirements on the coefficients: that $A$, $\mathbf{b}$ and $c$ be bounded on $\Omega$, which gives continuity with an $M$ built from those bounds, and that $A$ be uniformly elliptic, which is what replaces the identity matrix of the Poisson problem in the argument and gives coercivity with the same constant $\alpha$ already fixed by uniform ellipticity, provided the further condition below on the lower-order terms also holds. When convection is present ($\mathbf{b} \neq 0$) a further condition is needed, and a sufficient one is that $c - \tfrac{1}{2}\nabla \cdot \mathbf{b} \geq 0$ almost everywhere in $\Omega$, together with $\mathbf{b} \cdot \mathbf{n} \geq 0$ on $\Gamma_N$. The first condition holds automatically in the common case of a divergence-free convection field ($\nabla \cdot \mathbf{b} = 0$), where it reduces to asking that the reaction coefficient be non-negative ($c \geq 0$); the second says that the flow leaves the domain through the Neumann boundary rather than entering through it, and is vacuous for a pure Dirichlet problem. Note also that $a$ is symmetric only when $\mathbf{b} = 0$ and $A$ is symmetric — a property that will matter for the structure of the linear systems to come. A full treatment is given in Brenner and Scott, The Mathematical Theory of Finite Element Methods (Springer, 2008), Chapter 5.

This formulation is the foundation on which the discrete approximation is built. The next post replaces the infinite-dimensional space $V$ with a finite-dimensional subspace and derives the linear system that must be solved.