Abstract
The Cell Discretization algorithm is applied in a straightforward manner to a few boundary value problems whose solutions are singular in one way or another. When attempts are made to solve problems of this sort with simple analytic basis functions (e.g. polynomials, trigonometric functions), the result is an erratic oscillation usually called overshoot, ringing or the Gibbs phenomenon. When the singularities are explicitly incorporated into the representation of the solution, the irregularity can be completely eliminated. The examples chosen for this study are: (1) a Dirichlet problem with a jump in the boundary data at a corner, (2) a convection-diffusion problem with a boundary-value jump at a corner, (3) a convection-diffusion problem with steep boundary layers, (4) a vibrating membrane problem with a re-entrant corner, (5) a problem with a change from Dirichlet to Neumann boundary conditions on the same boundary face (Motz's problem). Copious figures illustrate the results.
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