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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)
where n is the outward unit normal vector.
The unit interval is (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 20, 2015
Contact: William Mitchell,
Development status: Active Development