Integration by Parts

Resources

Messenger, Daniel A., and David M. Bortz. 2021. “Weak SINDy for Partial Differential Equations.” Journal of Computational Physics 443 (October):110525. https://doi.org/10.1016/J.JCP.2021.110525.

The Finite Element Method

Main Idea

For an integration involving two functions, we can interchange differentiation operators between them, as long as we account for boundary conditions.

Definition

$\int_{Ω} u D v d Ω = - \int_{Ω} v D u d Ω + \int_{Γ} u v d Γ$

This is an application of the Divergence Theorem (Gauss's Theorem), which is

\int_{Ω} D w d Ω = \int_{Γ} w d Γ,

and we take $w = u v$ , apply the product rule for $D$ , and split the integral based on the summation in the integrand. Often this is presented with, $D = \nabla \cdot$ , but as seen in Weak SINDy, the differential operator $D$ can be more arbitrary (but must still be first order in this formula).

For a derivative $D^{α}$ , multi-indices $α$ and $β$ with order $| α |$ and $| β |$ , if

[D^{β} v]_{Γ} = 0, \forall β \in {β : β_{j} - 1 \leq α_{j}}

then,

\begin{aligned} \int_{Ω} u D^{α} v d Ω & = (- 1)^{| α |} \int_{Ω} v D^{α} u d Ω . \end{aligned}

In other words, if $v$ and a sufficient number of its derivatives are $0$ along the boundary, then the derivative operator is easily interchanged, up to the difference of the sign.

Summation by Parts

For a linear Finite Difference scheme $Δ$ ,

\sum_{i = m}^{n} v_{i} \cdot Δ u_{i} = v_{n + 1} u_{n + 1} - v_{m} u_{m} - \sum_{i = m}^{n} Δ v_{i} \cdot u_{i} .

One can take for instance $Δ y_{i} = y_{i + 1} - y_{i}$ and show this by expanding the left side, adding a final $0 = v_{n + 1} u_{n + 1} - v_{n + 1} u_{n + 1}$ , grouping the like $u_{i}$ terms, and factoring these terms. The technique is similar to that of telescoping series. This is also known as Abel's Transformation.

Resources

Related

Main Idea

Summation by Parts