The General Form PDE
The General Form PDE interface provides a general interface for specifying and solving PDEs in the general form. This format is closely related to the conservation laws that govern many areas of physics. Assuming that you are working with a single dependent variable u, the general form reads:
(16-1)
where
Ω is the computational domain; the union of all domains
∂Ω is the domain boundary
n is the outward unit normal vector on ∂Ω
The first line (equation) of Equation 16-1 is the PDE, which must be satisfied in Ω. The second, third, and fourth equations are the boundary conditions, which must hold on ∂Ω. The second equation is a generalization of a Neumann boundary condition. The third equation is a general constraint, of which the Dirichlet boundary condition on the fourth line is a special case.
The terms Γ, f, g, q, R, and r are user-defined coefficients. They can be functions of the spatial coordinates, the solution u, time, the space and time derivatives of u (see PDE Interface Variables), as well as of other predefined and user-defined variables. The coefficients f, g, q, R, and r are scalar, whereas Γ is the flux vector.
In practical applications, Γ typically represents the flux of a conserved quantity such as heat, charge, mass, or momentum. This flux is usually related in some empirical way, via a material law, to the gradient of the dependent variable. Therefore, Γ is usually a vector whose components are functions of derivatives of the dependent variable. The flux vector can also contain terms that are proportional to a velocity field when there is convective transport of the conserved quantity present. The structure of Equation 16-1 implies that the normal component of Γ is continuous across any surface in the interior of the domain, Ω .
Boundary Conditions for the General Form PDE
In finite element terminology, the boundary condition on the second line of Equation 16-2, corresponding to a Neumann boundary condition, is called a natural boundary condition, because it does not occur explicitly in the weak form of the PDE problem. In the PDE interfaces, the corresponding condition is called a flux or source, because it specifies the value of the numerical flux Γ at the boundary.
Constraints and Dirichlet conditions are also known as essential boundary conditions in finite element theory, because they impose a restriction on the trial space, which is not part of the main equation. In the PDE interfaces, a distinction is made between Dirichlet boundary conditions and constraints. The general constraint on line 3 of Equation 16-2 specifies that an arbitrary expression is equal to zero on the boundary: R = 0. The Dirichlet condition on line 4 of the same equation is a special case directly specifying the value of the dependent variable at the boundary: u = r. This makes the constraint a more general boundary condition.
The term hTμ in the generalized Neumann condition is a reaction term enforcing the constraint R = 0. When reaction terms are applied symmetrically on all dependent variables,
but other definitions are also possible. The variable μ is a Lagrange multiplier, which is eliminated by the solvers when using standard constraints and therefore does not normally appear explicitly in equations.