Logarithmic Norm - Applications To Elliptic Differential Operators

Applications To Elliptic Differential Operators

In connection with differential operators it is common to use inner products and integration by parts. In the simplest case we consider functions satisfying with inner product

Then it holds that

where the equality on the left represents integration by parts, and the inequality to the right is a Sobolev inequality. In the latter, equality is attained for the function, implying that the constant is the best possible. Thus

for the differential operator, which implies that

As an operator satisfying is called elliptic, the logarithmic norm quantifies the (strong) ellipticity of . Thus, if is strongly elliptic, then, and is invertible given proper data.

If a finite difference method is used to solve, the problem is replaced by an algebraic equation . The matrix will typically inherit the ellipticity, i.e., showing that is positive definite and therefore invertible.

These results carry over to the Poisson equation as well as to other numerical methods such as the Finite element method.