4. 3 Boundaries and Interfaces

From the line-wise point of view it is easy to define an equation for a boundary vertex. If, for instance, Dirichlet boundary conditions are employed, a respective residuum is formulated and a new equation for the vertex is inserted into the system matrix.

Especially element-wise formulation methods have difficulties with the specification of boundary conditions, because a special treatment for a boundary equation in a distinct vertex has to be considered in the calculation of all sub-matrices which are incident to the given vertex.

As a consequence, many equations for boundary points are treated algebraically which causes severe difficulties when introducing non-conventional boundary conditions, interface conditions, or conditions at triple points.

Every differential equation has to be specified appropriate boundary conditions in order to yield a valid solution. Without specifying a boundary condition, differential equations do not deliver a unique solution but offer a large function space of possible solutions.

In some cases it can be appropriate to have different sections, where interface conditions are given. This often turns out to be problematic for element-wise implementation approaches and various efforts have been spent on workarounds to handle this problem [83].