Volume 10, Issue 4

A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimalorder estimate for the point-wise error and a corresponding a posteriori error bound. The proofs are based on weighted L 2 -norm error estimates for discrete Green functions as already known for the ‘continuous’ finite element method.

A Galerkin finite element method that uses piecewise linear functions on Shishkin- and Bakhvalov–Shishkin-type of meshes is applied to a linear reaction-diffusion equation with discontinuous source term. The method is shown to be convergent, uniformly in the perturbation parameter, of order N –2 ln 2 N for the Shishkin-type mesh and N –2 for the Bakhvalov–Shishkin-type mesh, where N is the mesh size number. Numerical experiments support our theoretical results.

Important algorithms and design decisions of a new code, ode15i, for solving 0 = F ( t, y ( t ), y ′( t )) are presented. They were developed to exploit Matlab, a popular problem solving environment (PSE). Codes widely-used in general scientific computation (GSC) compute consistent initial conditions only for restricted forms of the differential equations. ode15i does this for the general problem, a task that is qualitatively different. Unlike popular codes in GSC, ode15i saves partial derivatives because this is more efficient in the PSE. A new representation of fixed leading coefficient BDFs is based on the Lagrangian form of polynomial interpolation. Basic computations are both clear and efficient in the PSE when using this form. Some numerical experiments show that ode15i is an effective way to solve fully implicit ODEs and DAEs of index 1 in Matlab.