?

open set in ⁿ

?

= ??

? _D

part of the boundary on which Dirichlet conditions are prescribed

? _N

part of the boundary on which Neumann conditions are prescribed

?

Laplace operator

L

differential operator

a _ik , a ₀

coefficient functions of the differential equation

[ ] _*

difference star, stencil

L ²( ?)

space of square-integrable functions over ?

H ^m( ?)

Sobolev space of L ₂ functions with square-integrable derivatives up to order m

( ?)

subspace of H ^m( ?) of functions with generalized zero bounary conditions

C ^k( ?)

set of functions with continuous derivatives up to order k

( ?)

subspace of C ^k( ?) of functions with compact support

?

trace operator

_m

Sobolev norm of order m

_m

Sobolev semi-norm of order m

_?

supremum norm

? ₂

space of square-summable sequences

H ?

dual space of H

,

dual pairing

?

= ? ? _i, order of multiindex ?

? _i

partial derivative

? ^?

partial derivative of order ?

D

(Fr chet) derivative

?

ellipticity constant

?, n

exterior normal

? _? , ?