Date: March 21
Topics Covered:
References: Johnson; Brenner & Scott
In the previous lecture, we developed the abstract Lax-Milgram theory, which provides existence, uniqueness, and stability for elliptic variational problems. Today, we put this theory into practice by analyzing several important PDEs from physics and engineering. We'll see how checking continuity and coercivity of the bilinear form $a(u,v)$ immediately guarantees well-posedness and convergence—this is the power of abstraction.
We begin with the biharmonic equation, which governs phenomena from acoustics to beam bending. Then we tackle reaction-diffusion equations that model chemical reactions, population dynamics, and pattern formation. Most importantly, we'll examine linear elasticity, where a fascinating challenge emerges: as materials become nearly incompressible (like rubber or hydrogels), the Poisson ratio $\nu \to 1/2$ causes the Lamé parameter $\lambda \to \infty$, and our standard Galerkin formulation loses stability.
This breakdown motivates mixed finite element methods, where we introduce additional variables (like pressure or stress) and solve for multiple coupled fields simultaneously. The key mathematical tool is the inf-sup condition (also called Ladyzhenskaya-Babuška-Brezzi or LBB condition), which prescribes precise compatibility requirements between the function spaces for different fields. We'll see this framework emerge naturally from the Stokes equations for incompressible fluid flow, which is the limiting case of nearly incompressible elasticity. Understanding when and why standard FEM fails, and how mixed methods rescue us, is crucial for building robust physics-informed learning algorithms.
Final Exam: April 30th
Final Project: Plan projects now; presentations May 5-13
Last time we established the abstract framework:
(G) Find $u \in V_h$ where $V_h \subseteq V$ such that:
$$a(u,v) = L(v) \quad \forall v \in V_h$$Lax-Milgram conditions:
(1) Continuity: $a$ is continuous if
$$|a(v,w)| \leq \gamma \|v\|_V \|w\|_V$$(2) Ellipticity (coercivity): $a$ is elliptic if
$$a(v,v) \geq \alpha \|v\|_V^2$$(3) Bounded linear functional: $L$ is continuous if
$$|L(v)| \leq \Lambda \|v\|_V$$If (1)-(3) hold, then:
Remember:
Choose: $V = H_0^1(\Omega)$ (functions with square-integrable first derivatives, zero on boundary)
Inner product:
$$(u,v)_V = \int_\Omega \nabla u \cdot \nabla v \, dx$$(1) Continuity:
$$\begin{aligned} |a(v,w)| &= \left|\int \nabla v \cdot \nabla w \, dx\right| \\ &\leq \|\nabla v\|_{L^2} \|\nabla w\|_{L^2} \quad \text{(Cauchy-Schwarz)} \\ &\leq \|v\|_V \|w\|_V \end{aligned}$$(2) Ellipticity:
$$a(v,v) = \int |\nabla v|^2 \, dx = |v|_{H^1}^2$$For $v \in H_0^1$, by Poincaré inequality: $\|v\|_{L^2} \leq C |v|_{H^1}$
Therefore: $a(v,v) \geq \alpha \|v\|_{H^1}^2$ for some $\alpha > 0$.
(3) Bounded functional:
$$|L(v)| \leq \|f\|_{L^2} \|v\|_{L^2} \leq \|f\|_{L^2} \|v\|_V$$All conditions satisfied! ✓
The biharmonic equation arises in:
(Both function and normal derivative vanish on boundary—clamped plate conditions)
(Functions with square-integrable second derivatives)
(1) Continuity: Follows from Cauchy-Schwarz
(2) Ellipticity: This is non-trivial—requires a result from PDEs called elliptic regularity:
Elliptic regularity theorem: For convex $\Omega$,
$$\|v\|_{H^2(\Omega)} \leq C \|\Delta v\|_{L^2(\Omega)} \quad \forall v \in H^2 \cap H_0^1$$This ensures $a(v,v) \geq \alpha \|v\|_{H^2}^2$.
(3) Bounded functional:
$$|L(v)| \leq \|f\|_{L^2} \|v\|_{L^2} \leq \|f\|_{L^2} \|v\|_V$$All conditions satisfied! ✓
with assumptions:
where $v \in H_0^1(\Omega)$.
(1) Continuity:
Diffusion term:
$$\begin{aligned} \left|\int \nabla v \cdot A \cdot \nabla w \, dx\right| &\leq \max_{i,j} |A_{ij}| \left|\int \nabla v \cdot \nabla w \, dx\right| \\ &\leq \gamma_A |v|_{H^1} |w|_{H^1} \end{aligned}$$Reaction term:
$$\left|\int r v w \, dx\right| \leq \max|r| \int |vw| \, dx \leq \gamma_r \|v\|_{L^2} \|w\|_{L^2}$$Combined:
$$|a(v,w)| \leq \gamma_A |v|_{H^1} |w|_{H^1} + \gamma_r \|v\|_{L^2} \|w\|_{L^2} \leq 2\max(\gamma_A, \gamma_r) \|v\|_{H^1} \|w\|_{H^1}$$(2) Ellipticity:
$$\begin{aligned} a(u,u) &= \int \nabla u \cdot A \cdot \nabla u + ru^2 \, dx \\ &\geq \int |\nabla u|^2 + u^2 \, dx \quad \text{(by positive definiteness and } r \geq 0\text{)} \\ &= \|u\|_{H^1}^2 \end{aligned}$$(3) Bounded functional: Same as Poisson.
All conditions satisfied! ✓
Displacement: $u \in \mathbb{R}^d$
Strain tensor:
$$\varepsilon(u) = \frac{1}{2}(\nabla u + \nabla u^\top) \in \mathbb{R}^{d \times d}$$(symmetric part of displacement gradient)
Stress tensor: $\sigma(\varepsilon)$
Constitutive law (Hooke's law):
$$\sigma_{ij} = C_{ijkl} \varepsilon_{kl}$$To enforce rotational invariance for isotropic materials:
$$C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})$$where:
The stress reduces to:
$$\boxed{\sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk} + 2\mu \varepsilon_{ij}}$$or in vector notation:
$$\sigma = 2\mu \varepsilon + \lambda \operatorname{tr}(\varepsilon) I$$Note: $\operatorname{tr}(\varepsilon) = \nabla \cdot u$ (volumetric strain)
From variational mechanics, the Lagrangian density for elasticity is:
$$\mathcal{L} = -\frac{1}{2} \varepsilon^\top C \varepsilon + \frac{\rho}{2} (\partial_t u)^2$$Combining:
$$\begin{cases} -\nabla \cdot (2\mu \varepsilon(u) + \lambda \nabla \cdot u \, I) = f \\ u|_{\partial\Omega} = 0 \end{cases}$$Work with displacements in vector space:
$$\vec{u} \in \vec{H}_0^1(\Omega) = [H_0^1(\Omega)]^d$$(i.e., each component is in $H_0^1$)
$$\boxed{a(u,v) = \int_\Omega \mu \varepsilon(u):\varepsilon(v) + \lambda(\nabla \cdot u)(\nabla \cdot v) \, dx}$$where $A:B = \sum_{ij} A_{ij} B_{ij}$ is the Frobenius inner product.
The Lamé parameters relate to engineering constants via:
$$\lambda = \frac{2\mu \nu}{1 - 2\nu}$$where $\nu$ is the Poisson ratio:
$$\nu = -\frac{\varepsilon_{\text{transverse}}}{\varepsilon_{\text{axial}}}$$Problem: For near-incompressible materials (e.g., rubber, hydrogels, clays, saturated porous media):
$$\nu \to \frac{1}{2} \quad \Rightarrow \quad \lambda \to \infty$$We need to understand the $\lambda \to \infty$ limiting stability!
To analyze this, we need Korn's inequality:
$$\boxed{\int \varepsilon(u):\varepsilon(u) \, dx \geq C \|u\|_{H^1}^2}$$This says the strain energy controls the $H^1$ norm of displacement.
(1) Continuity:
Strain energy term:
$$\begin{aligned} \left|\int \mu \varepsilon(w):\varepsilon(v) \, dx\right| &= \left|\frac{\mu}{4} \sum_{i,j} \int (\partial_{x_i} w_j + \partial_{x_j} w_i)(\partial_{x_i} v_j + \partial_{x_j} v_i) \, dx\right| \\ &= \mu \sum_{i,j} \int |\partial_{x_i} w_j|^2 \, dx \\ &\leq \mu \|w\|_{H^1} \|v\|_{H^1} \end{aligned}$$Volumetric term:
$$\left|\int \lambda (\nabla \cdot w)(\nabla \cdot v) \, dx\right| \leq \lambda \|w\|_{H^1} \|v\|_{H^1}$$Combined:
$$|a(w,v)| \leq (\mu + \lambda) \|w\|_{H^1} \|v\|_{H^1}$$(2) Ellipticity:
$$\begin{aligned} a(u,u) &= \int \mu \varepsilon(u):\varepsilon(u) + \lambda(\nabla \cdot u)^2 \, dx \\ &\geq \int \mu \varepsilon(u):\varepsilon(u) \, dx \\ &\geq \mu \|u\|_{H^1}^2 \quad \text{(by Korn)} \end{aligned}$$From Lax-Milgram:
(1) Stability: $\|u_h\|_{H^1} \leq \Lambda/\mu$ → Get a stable solution
(2) By Céa's lemma:
$$\|u - u_h\|_{H^1} \leq \frac{\mu + \lambda}{\mu} \|u - v\|_{H^1}$$Problem: As $\lambda \to \infty$, the constant $\frac{\mu + \lambda}{\mu} \to \infty$!
The error bound blows up, even though a solution exists.
This is called locking in finite element analysis—the discrete solution loses accuracy as the material becomes incompressible.
One solution to locking is mixed finite element methods.
Take the original form:
$$\mu \int \varepsilon(u):\varepsilon(v) \, dx + \lambda \int (\operatorname{div} u)(\operatorname{div} v) \, dx = \int f \cdot v \, dx$$Introduce new variable: Let $p = \lambda \operatorname{div} u$ (essentially a pressure).
Introduce a second FEM space $q \in M_h$.
This gives a new Galerkin equation, but how do we choose $M_h$?
Note: In the limit $\mu = 1$, $\lambda \to \infty$, this reduces to the stationary Stokes problem:
$$(S) \quad \begin{cases} -\Delta u - \nabla p = f \\ \nabla \cdot u = 0 \end{cases}$$where:
Physical interpretation: Incompressible viscous fluid flow.
By integration by parts:
$$(\nabla p, v) = -(p, \nabla \cdot v) \leq \|p\|_{L^2} \|v\|_{H^1}$$Galerkin form:
$$\begin{cases} (\nabla u, \nabla v) + (p, \nabla \cdot v) = (f, v) & \forall v \in V_h \\ (\nabla \cdot u, q) = 0 & \forall q \in M_h \end{cases}$$This is a saddle-point problem (not a minimization).
We can prove uniqueness quickly for the case $f = 0 \Rightarrow p, u = 0$.
Step 1: Take $v = u$ in momentum equation:
$$\|u\|_{H^1}^2 + (p, \nabla \cdot u) = 0$$Step 2: Take $q = p$ in continuity equation:
$$(\nabla \cdot u, p) = 0$$Step 3: Substitute into Step 1:
$$\|u\|_{H^1}^2 = 0 \quad \Rightarrow \quad u = 0$$Problem: We'd like to show $u = 0 \Rightarrow p = 0$, but we can't!
Nothing in the equations directly gives us information about $p$ when $u = 0$.
Key idea: Design a relationship between $V_h$ and $M_h$ so that $u = 0 \Rightarrow p = 0$.
$V_h$ and $M_h$ are inf-sup compatible if:
For any $q \in M_h$, there exists $v \in V_h$ such that:
$$\boxed{\beta \|q\|_{L^2} \leq \frac{(q, \nabla \cdot v)}{\|v\|_{H^1}}}$$for some constant $\beta > 0$ independent of $h$.
This is also called:
Return to Stokes momentum equation with $f = 0$:
$$(\nabla u, \nabla v) + (p, \nabla \cdot v) = 0$$We already showed $u = 0$.
Take $q = p$ and use inf-sup:
$$\|p\|_{L^2} \leq \frac{(p, \nabla \cdot v)}{\beta \|v\|_{H^1}} = \frac{0}{\beta \|v\|_{H^1}} = 0$$Therefore: $p = 0$ ✓
The inf-sup condition ensures that pressure can be uniquely determined from velocity.
Without it, the pressure is under-determined (non-unique), leading to spurious pressure modes and numerical instabilities.
Geometrically: It says that the divergence operator from $V_h$ to $M_h$ has a bounded inverse (in a weak sense).
This lecture covered:
Key Takeaway: Not all PDEs can be solved with standard Galerkin FEM. When dealing with near-incompressibility (elasticity, Stokes flow), or more generally when solving for multiple coupled fields with different regularity requirements, we need mixed finite element methods. The inf-sup condition provides the mathematical criterion for choosing compatible function spaces $V_h$ and $M_h$—violate it, and you get spurious modes, locking, or non-unique solutions. Understanding this framework is essential for physics-informed machine learning: when learning operators or spaces for coupled multiphysics problems, we must preserve the algebraic structure (saddle-point form) and ensure learned spaces satisfy inf-sup compatibility to guarantee well-posed discrete problems.
Next time: How to build inf-sup stable spaces in practice; How to learn physics in the mixed FEM setting