Weak / variational formulation of elliptic PDE

Intuition Beginner

A weak formulation rewrites a differential equation as an energy-balance rule.

Instead of asking the solution to satisfy a formula at every point, we ask it to satisfy a weighted average identity against many test functions. This allows solutions with limited smoothness and makes boundary conditions part of the function space.

For elliptic PDEs, the weak form often says: find the function whose energy pairing with every allowed test function matches the forcing.

This is exactly what finite elements need. Once the problem is written as a weak form, we can replace the infinite-dimensional space by a finite-dimensional trial space and solve a linear algebra problem.

Visual Beginner

The test functions probe the equation in an averaged way. The mesh version uses a finite set of such probes.

Worked example Beginner

Picture a stretched membrane pushed by a load. A pointwise equation describes the balance at each location, but the weak version asks for balance against every allowed deformation of the membrane.

If a trial deformation lowers the energy, the membrane was not in equilibrium. At the solution, every allowed small deformation has the correct energy balance against the applied load.

What this tells us: weak solutions are not weaker because they are less useful. They are weaker because they require less pointwise smoothness while keeping the physically meaningful balance law.

Check your understanding Beginner

Formal definition Intermediate+

Let $V$ be a Hilbert space, let $a:V\times V\to \mathbb{R}$ be a bilinear form, and let $F:V\to \mathbb{R}$ be a continuous linear functional. A weak variational problem asks: $$ \text{find }u\in V\text{ such that }a(u,v)=F(v)\quad\text{for all }v\in V. $$

For the model Dirichlet Poisson problem $$ -\Delta u=f\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega, $$ the weak space is $V=H_0^1(\Omega )$ and the weak form is $$ \int_\Omega \nabla u\cdot\nabla v,dx=\int_\Omega f v,dx \quad\text{for all }v\in H^1_0(\Omega). $$

The bilinear form is continuous if $$ |a(u,v)|\leq M|u|_V|v|_V, $$ and coercive if $$ a(v,v)\geq \alpha|v|_V^2 $$ for some $\alpha >0$ and all $v\in V$.

The Galerkin approximation chooses a finite-dimensional subspace $V_h\subset V$ and asks: $$ \text{find }u_h\in V_h\text{ such that }a(u_h,v_h)=F(v_h) \quad\text{for all }v_h\in V_h. $$

Counterexamples to common slips

• A weak formulation is not a numerical method by itself. It becomes Galerkin/FEM after choosing a finite-dimensional space.
• Coercivity is stronger than continuity. Continuity bounds the bilinear form above; coercivity bounds the energy below.
• Boundary conditions are not forgotten. Essential boundary conditions are often encoded in the choice of $V$.

Key theorem with proof Intermediate+

Theorem (Lax-Milgram). Let $V$ be a Hilbert space. If $a:V\times V\to \mathbb{R}$ is continuous and coercive, and $F\in V^{\ast }$ is continuous, then there is a unique $u\in V$ such that $$ a(u,v)=F(v)\quad\text{for all }v\in V. $$

Proof. For each fixed $u\in V$, the map $v\mapsto a(u,v)$ is a continuous linear functional on $V$. By the Riesz representation theorem, there is a unique element $Au\in V$ such that $$ a(u,v)=\langle Au,v\rangle_V $$ for all $v\in V$. This defines a bounded linear operator $A:V\to V$.

Coercivity gives $$ \langle Av,v\rangle_V=a(v,v)\geq \alpha|v|_V^2. $$ This implies injectivity and a lower bound $\Vert Av{\Vert }_V\ge \alpha \Vert v{\Vert }_V$. Hence the range of $A$ is closed. The range is also dense: if $w$ is orthogonal to the range, then $⟨Av,w⟩=0$ for all $v$, so in particular the adjoint relation gives $A^{\ast }w=0$; coercivity applied through the adjoint form forces $w=0$. Thus the range is all of $V$.

By Riesz again, $F(v)=⟨f,v{⟩}_V$ for a unique $f\in V$. Since $A$ is onto, choose $u$ with $Au=f$. Then $$ a(u,v)=\langle Au,v\rangle_V=\langle f,v\rangle_V=F(v). $$ Uniqueness follows from coercivity applied to the difference of two solutions. $\square$

Bridge. Weak formulations use the Sobolev spaces of 24.01.01 as solution and test spaces. The Lax-Milgram theorem converts continuity and coercivity into existence and uniqueness. This directly supports 24.02.01, where Galerkin finite elements approximate the weak problem, and 24.01.04, where saddle-point problems require inf-sup stability instead of coercivity on one space.

Exercises Intermediate+

Advanced results Master

The variational formulation of elliptic PDE is the point where analysis becomes computation. Once a PDE is expressed as $a(u,v)=F(v)$ on a Hilbert space, finite-dimensional approximation is obtained by replacing $V$ with a subspace $V_h$.

For the Poisson equation, coercivity comes from the gradient energy and the Poincare or Friedrichs inequality. On $H_0^1(\Omega )$, $$ |u|{H^1}\leq C|\nabla u|{L^2}, $$ so the bilinear form $a(u,v)={\int }_{\Omega }\nabla u\cdot \nabla v$ controls the full $H^1$ norm.

The Galerkin orthogonality relation $$ a(u-u_h,v_h)=0\quad\text{for all }v_h\in V_h $$ is the algebraic heart of finite element error analysis. It is the reason Céa's lemma can compare the actual error with the best approximation error in the chosen finite element space.

Weak formulations also clarify boundary conditions. Essential boundary conditions are built into the trial space, while natural boundary conditions arise from integration by parts and appear in the linear functional or boundary terms.

Not every elliptic problem is coercive on a single Hilbert space. Mixed formulations introduce multiple unknowns and a saddle-point structure. Those problems require the Babuška-Brezzi inf-sup condition in 24.01.04, which generalizes the stability role played here by coercivity.

Synthesis. A weak formulation turns a PDE into a Hilbert-space equation whose stability can be checked by functional analysis. Lax-Milgram gives well-posedness for coercive problems; Galerkin restriction gives finite-dimensional approximations; Galerkin orthogonality gives error estimates. This is the pipeline from Sobolev spaces to classical finite element methods.

Full proof set Master

Proposition 1 (Poisson coercivity). On $H_0^1(\Omega )$, the bilinear form $$ a(u,v)=\int_\Omega\nabla u\cdot\nabla v,dx $$ is coercive with respect to the $H^1$ norm on a bounded regular domain.

Proof. The Poincare inequality gives $\Vert u{\Vert }_{L^2}\le C\Vert \nabla u{\Vert }_{L^2}$. Hence $$ |u|{H^1}^2=|u|{L^2}^2+|\nabla u|{L^2}^2 \leq (C^2+1)|\nabla u|{L^2}^2. $$ Since $a(u,u)=\Vert \nabla u{\Vert }_{L^2}^2$, this gives $$ a(u,u)\geq (C^2+1)^{-1}|u|_{H^1}^2. $$ $\square$

Proposition 2 (Galerkin orthogonality). If $u$ solves the continuous weak problem and $u_h$ solves the Galerkin problem in $V_h\subset V$, then $$ a(u-u_h,v_h)=0 $$ for every $v_h\in V_h$.

Proof. Since $u$ solves the continuous problem, $a(u,v_h)=F(v_h)$ for every $v_h\in V_h$. Since $u_h$ solves the discrete problem, $a(u_h,v_h)=F(v_h)$ for every $v_h\in V_h$. Subtracting gives the result. $\square$

Proposition 3 (energy best approximation in the symmetric case). If $a$ is symmetric, continuous, and coercive, then the Galerkin solution minimizes the energy norm of the error over $V_h$.

Proof. Define $\Vert w{\Vert }_a^2=a(w,w)$. For any $v_h\in V_h$, $$ u-v_h=(u-u_h)+(u_h-v_h). $$ The second term lies in $V_h$, so Galerkin orthogonality gives $$ a(u-u_h,u_h-v_h)=0. $$ Therefore $$ |u-v_h|_a^2=|u-u_h|_a^2+|u_h-v_h|_a^2\geq|u-u_h|_a^2. $$ Thus $u_h$ is the best approximation in the energy norm. $\square$

Connections Master

• Sobolev spaces 24.01.01. Weak formulations use Sobolev spaces as trial and test spaces.

• Babuška-Brezzi condition 24.01.04. Saddle-point weak formulations need inf-sup stability rather than simple coercivity.

• Classical conforming FEM 24.02.01. Galerkin restriction is the analytic foundation of conforming finite elements.

• Mixed FEM for Poisson 24.02.02. Mixed weak forms introduce flux variables and saddle-point structure.

• FEEC convergence theorem 24.03.06. FEEC generalizes Galerkin stability from scalar elliptic PDE to Hilbert complexes.

Historical & philosophical context Master

Galerkin's 1915 method approximated solutions by finite linear combinations and enforced residual orthogonality against chosen test functions ^[Galerkin]. This idea later became one of the central templates for finite element methods.

The Lax-Milgram theorem supplied a clean functional-analytic existence theorem for coercive variational problems ^{[Lax-Milgram]}. It explains why many elliptic boundary-value problems are well-posed at the weak level.

Modern finite element analysis combines these two lines: first prove the weak problem is stable in a Sobolev space, then choose a finite-dimensional subspace and analyze the Galerkin approximation ^{[Brenner-Scott]}. This is the pattern that later FEEC extends from scalar functions to differential complexes.

Bibliography Master

@article{Galerkin1915,
  author = {Galerkin, Boris G.},
  title = {Series solution of some cases of equilibrium of elastic beams and plates},
  journal = {Vestnik Inzhenerov i Tekhnikov},
  volume = {1},
  pages = {897--908},
  year = {1915}
}

@incollection{LaxMilgram1954,
  author = {Lax, Peter D. and Milgram, Arthur N.},
  title = {Parabolic equations},
  booktitle = {Contributions to the Theory of Partial Differential Equations},
  series = {Annals of Mathematics Studies},
  volume = {33},
  pages = {167--190},
  publisher = {Princeton University Press},
  year = {1954}
}

@book{Evans2010WeakPDE,
  author = {Evans, Lawrence C.},
  title = {Partial Differential Equations},
  edition = {2},
  series = {Graduate Studies in Mathematics},
  volume = {19},
  publisher = {American Mathematical Society},
  year = {2010}
}

@book{BrennerScott2008WeakFEM,
  author = {Brenner, Susanne C. and Scott, L. Ridgway},
  title = {The Mathematical Theory of Finite Element Methods},
  edition = {3},
  publisher = {Springer},
  year = {2008}
}