From Equation to Solution

This is a continuation of the previous post on the role of functional analysis, more precisely the role of the finite element method as a form of computational functional analysis.

We start with the basic partial differential equation of physics and mechanics, Poisson's equation:

• $-\Delta u(x) = f(x)$ for $x\in\Omega$,
• $u(x)=0$ for  $x\in\Gamma$, 

where $\Omega$ is a domain in space with boundary $\Gamma$, $f(x)$ is a given function defined on $\Omega$ and $u(x)$ is the solution to the equation defined on $\Omega$ and $\Gamma$. The game is: Given $f(x)$ find $u(x)$ satisfying Poisson's equation.

We can think of the differential equation $-\Delta u(x)=f(x)$ as expressing force balance at the point $x$ with $u(x)$ the deflection of an elastic membrane under a transversal force or load $f(x)$, in case $\Omega$ is two-dimensional.  There are endless other interpretations.

So far so good, the partial differential equation $-\Delta u=f$ captures complex physics in very compact beautiful mathematical form, and so is marvellous, but there is one caveat: The formulation of the equation gives no clue to how to determine the solution $u(x)$. The equation is like a rebus without any hint of resolution.

It is here that functional analysis enters by offering a reformulation of the differential equation $-\Delta u =f$ into variational form: Find $u\in V$ such that

• $\int_\Omega \nabla u\cdot\nabla v\, dx = \int_\Omega fv\, dx$ for all $v\in V$,       (1) 

where $V$ is a collection (function space) of possible solutions, from which a best possible solution $u(x)$ is determined by the relation (1). Formally (1) is obtained by multiplying the differential equation $-\Delta u=f$ on both sides with an arbitrary function $v\in V$ and integrating over $\Omega$ using integration by parts to see that (using that $v=0$ on $\Omega$)  

• $-\int\Delta uv\, dx =\int_\Omega\nabla u\cdot\nabla v\, dx$. 

In the finite element method the space $V$ consists of piecewise polynomial functions over a triangulation of $\Omega$ and (1) is a linear system of algebraic equations, which can be solved by Jacobi iteration or Gaussian elimination. 

The differential equation as unsolvable rebus has thus been reformulated into variational form which allows a best possible solution to be computed by standard linear algebra software.  Here functional analysis enters in the variational formulation and the construction of the finite element space $V$.

The great thing is now that the same method works for virtually any (partial) differential equation, in particular the differential equations of science and technology: Reformulating the differential equation into variational form allows computation of best possible (approximate) solution. 

This is realised in the FEniCS Project which is software automating the whole process consisting of 

• reformulation into variational form, 
• construction of finite element space $V$,
• computation of solution by linear algebra. 

The crown jewel is automated computation of best possible solution of Navier-Stokes equations which we claim resolves the Clay Navier-Stokes Problem and makes turbulent flow computable and thus understandable, for the first time. And this is only the beginning of a FEniCS revolution.

We understand that the differential equation $-\Delta u(x)=f(x)$ expresses local force balance (at the point x), while the solution $u(x)$ comes out as a global effect depending on $f(y)$ for all $y$ and not just $f(x)$. This means that to determine $u(x)$ requires computation collecting many local inputs to one global output.

The mathematics of Jacobi iteration then corresponds to the physics of relaxation where the system reacts to reduce force imbalance. Gaussian elimination (or even better multi-grid) is more efficient than Jacobi iteration, which allows mathematics to take a short-cut to solution compared to physical relaxation.

PS The Navier-Stokes-Euler equations for incompressible flow contains the equation

• $\nabla\cdot u=0$ 

expressing the incompressibility, together with an equation expressing force balance according to Newton's 2nd law. The equation $\nabla\cdot u=0$ does not express force balance and appears more like a regulation stipulating a certain property of the solution (incompressibility) than a true law of physics like Newton's 2nd law. In DFS (near) incompressibility is instead expressed as a pressure law of basic form

• $\Delta p=\frac{\nabla\cdot u}{\delta}$  

where $\delta > 0$ is a small parameter, with the effect of forcing $\nabla\cdot u$ to be small by pressure as an expression of some physics. The lesson is that a differential equation without solution procedure is only half of the story.  Stating laws without means of enforcing the laws may be empty.