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A Petrov-Galerkin mixed finite element method with exponential fitting. (English) Zbl 0840.65130
The author is interested in the semiconductor continuity equation on a rectangular region with Dirichlet boundary conditions. The numerical methods are tied to rectangular mesh elements. The discretization is constructed by first expressing the problem using a mixed formulation, then approximating the coefficients as constants in each mesh element, and, finally introducing bilinear forms based on a special quadrature. This quadrature is designed so that the case of constant coefficients and exponential solution is exactly reproduced. A priori \(O(h)\) error estimates are demonstrated, an a posteriori error estimator (based on higher-order quadrature) is presented, and numerical results are presented. The numerical results show better convergence behavior than the theoretical estimates.
The author presents three error estimates, the first pair of estimates indicates that the error in the solution is \(O(h)\), for \(h\) small enough. The third relaxes the condition on \(h\) in the error estimate for the solution by employing estimates of the discrete adjoint problem.
The author observes that the special exponential weighting reduces to an upwind difference method in the convection-dominated limit. Further, use of this weighting results in an \(M\)-matrix to be inverted for the unknown, after static condensation is used to eliminate its gradient.

MSC:
65Z05 Applications to the sciences
35Q60 PDEs in connection with optics and electromagnetic theory
78A55 Technical applications of optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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