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3.1 Mathematical background

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To approximate a set of linear partial differential equations

$\displaystyle \mathbf{L}\vec{v}=\vec{r}\;\;\;\textnormal{in}\;\;\; \Omega$ (1)

for an unknown $ \vec{v}\in\mathcal{V}\subset\mathcal{C}^n(\Omega)$ that is continuously defined with $ n$  derivatives in the volume $ \Omega$ and is subject to the boundary conditions

$\displaystyle \mathbf{B}\vec{v}=\vec{s}\;\;\;\textnormal{in}\;\;\; \partial\Omega$ (2)

a mathematician involves first the so-called

* Weighted residuals.
Having defined a scalar product $ (\cdot, \cdot)$ and a norm $ \vert\vert\cdot\vert\vert$ , the calculation essentially amounts to the minimization of a residual vector

$\displaystyle \vert\vert\vec{R}\vert\vert=\vert\vert\vec{r}-\mathbf{L}\vec{v}\vert\vert$ (3)

which is carried out using tools from the variational calculus.

* Variational principle.
A quadratic form is constructed for that purpose by choosing a test function $ \vec{w}$ in a sub-space $ \mathcal{W}$ that is ``sufficiently general'' and satisfies the boundary conditions. The linear equation (3.1#eq.1) can then be written as an equivalent variational problem

$\displaystyle (\vec{w},\vec{R})=(\vec{w},\mathbf{L}\vec{v}-\vec{r})=0 \hspace{10mm} \vec{v}\in\mathcal{V},\;\;\forall \vec{w}\in\mathcal{W}$ (4)

* Integration by parts.
If $ \vec{w}$ is differentiable at least once in $ \mathcal{W}\subset\mathcal{C}^1(\Omega)$ , the regularity required by the forth-coming discretization can often be relaxed by partial integrations. Using $ \mathcal{L}=\nabla^2$ for illustration, Leibniz' rule states that

$\displaystyle \nabla\cdot(v \vec{w}) = (\nabla v)\cdot\vec{w} +v\nabla\cdot\vec... ...arrow\;\;\; \nabla v\cdot\vec{w} = -v\nabla\cdot\vec{w} +\nabla\cdot(v \vec{w})$ (5)

Integrating over the volume $ \Omega$ and using Gauss' divergence theorem

$\displaystyle \int_\Omega \nabla\cdot\vec{F}\;dV= \int_{\partial\Omega} \vec{F}\cdot\vec{dS}$    

yields a generalized formula for partial integration:

$\displaystyle \int_\Omega \nabla v\cdot\vec{w}\;dV = -\int_\Omega v\nabla\cdot\vec{w}\;dV +\int_{\partial\Omega} v\vec{w}\cdot\vec{dS}$ (6)

For the special case where $ \vec{w}=\nabla u$ , this is known as Green's formula

$\displaystyle \int_\Omega v\nabla^2 u\;dV= -\int_\Omega \nabla v\cdot\nabla u \;dV -\int_{\partial\Omega} v\nabla u\cdot\vec{dS}$ (7)

The last (surface-) term can sometimes be imposed to zero (or to a finite value) when applying so-called natural boundary conditions.

* Numerical approximation.
It turns out that the formulation as a variational problem is general enough that the solution $ \vec{v}$ of (3.1#eq.4) remains a converging approximation of (3.1#eq.1) even when the sub-spaces $ \mathcal{V,W}$ are restricted to finite, a priori non-orthogonal, but still complete sets $ \overline{\mathcal{V}}, \overline{\mathcal{W}}$ of functions. The overlap integrals involving non-orthogonal functions are often evaluated with a numerical quadrature that can be easily be handled by the computer.

In general, the discretized solution $ \vec{v}$ is expanded in basis functions $ \vec{e_j}\in\overline{\mathcal{V}}\subset\mathcal{V}$ , which either reflect a property of the solution (e.g. the operator Green's function in the Method of Moments), or which are simple and localized enough so that they yields cheap inner products $ (\vec{w}, \vec{e_j})$ and sparse linear systems (e.g. the roof-top function for the linear Finite Element Method). Different choices can also be made for the test functions $ \vec{w}$ : among the most popular are the Galerkin method, where test and basis functions are both chosen from the same sub-space $ \overline{\mathcal{V}}\equiv\overline{\mathcal{W}}$ and the method of collocation, which consists in taking Dirac functions $ \vec{w}\in\overline{\mathcal{W}} = \{\delta(\vec{x}-\vec{x_j})\}, j=1,N$ that lead to point-wise evaluations of the integrand on the mesh $ \{\vec{x_j}\}, j=1,N$ .

* Boundary conditions.
If natural boundary conditions are not already sufficient, essential boundary conditions have to be imposed either by choosing the functional space $ \overline{\mathcal{V}}, \overline{\mathcal{W}}$ (e.g. let $ \vec{e_1}$ and $ \vec{e_N}$ overlap in a periodic domain $ [x_1; x_n]$ as in the next section) or by replacing one or several equations of the linear system (e.g. use higher order finite differences to preserve the convergence rate as in exercise 3.02).

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