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Conventions, Notations and Terminology

For many problems with a known exact solution, the right hand side of the differential equation is given as f (or f1 or some other related form) and the right hand side of the boundary condition is given as g (or a related form). In this case the right hand side is determined by applying the operator to the exact solution.

Poisson's Equation (in 2D) is -uxx - uyy = f, with the obvious modifications for other numbers of dimensions.

Laplace's Equation is Poisson's Equations with f = 0.

The Dirichlet boundary condition is u = g.

The Neumann boundary condition is ∂u/∂n = g where n is the outward unit normal.

The natural boundary condition for (for example) -∇·(pu)=f is pu·n=g where n is the outward unit normal vector.
When p=1, the natural boundary conditions are Neumann.

The unit interval is (0,1).
The unit square is (0,1)X(0,1).
The unit cube is (0,1)X(0,1)X(0,1).

Semilinear equations of order k are those in which the coefficients of the kth order terms do not depend on u or its derivatives, but other terms may be nonlinear in u and derivatives of u up to order k-1.

Quasilinear equations of order k are those in which the coefficients of all terms, including the kth order terms, may depend on u and derivatives of u up to order k-1.

Fully nonlinear equations of order k are nonlinear in the kth derivatives of u.

A goal-oriented problem is one for which the objective is to minimize the error in some prescribed quantity of interest. For non-goal-oriented problems the objective is usually to minimize some global norm of the error.




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Date created: May 16, 2013 | Last updated: August 21, 2013    Contact: William Mitchell, Home Page
Development status: Active Development