3.1 Mathematical background

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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

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

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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