Mixed vector boundary conditions constrain components relative to the local boundary normal. At a node in the interior of a boundary face, that normal is unambiguous. At an edge or corner, however, several boundary faces can meet and may impose different kinds of constraint. This appendix defines how those face constraints are reduced to one homogeneous nodal constraint.

The construction acts on vectors used by a linear solve, such as corrections or residuals, for which the boundary constraints are homogeneous. Prescribed non-zero boundary values are outside the scope of this construction.

Boundary classes and priority

Each incident boundary face has one of four class values:

Class	Constraint	Admissible homogeneous vectors
0	All components	Only the zero vector
2	Normal component	Vectors tangent to the boundary
3	Tangential components	Vectors parallel to the boundary normal
5	None	All vectors

The numerical values define the priority order

$$0 \mathrel{\succ} 2 \mathrel{\succ} 3 \mathrel{\succ} 5,$$

where a smaller value has higher priority. This is a policy for resolving different boundary classes; it is not an ordering by the dimension of their constraint spaces.

Let $\mathcal{F}(x)$ be the set of boundary faces incident on a node $x$, and let $c_f$ denote the class of face $f$. The nodal class is

$$c(x) = \min_{f \in \mathcal{F}(x)} c_f.$$

Consequently, a fully constrained face takes priority over every other class, and a normal-component constraint takes priority over a tangential-component constraint.

Resolution of the nodal normal

A local normal is required only when $c(x)$ is class 2 or class 3. Among the faces incident on $x$, only those whose class equals the resolved nodal class contribute. Define

$$ \mathcal{F}_c(x) = \left\{ f \in \mathcal{F}(x) : c_f = c(x) \right

Table of Contents

Boundary classes and priority

Resolution of the nodal normal