Abstract
A boundary value problem for the Lavrent’ev–Bitsadze differential equation of mixed type is divided into an elliptic problem involving an unusual boundary condition on the parabolic line, and a hyperbolic problem. The elliptic problem is then converted into an equivalent variational form and shown to be well posed under certain conditions which always include the characteristic boundary case. The finite element method is applied using triangular elements and a few singular functions near the intersection of the parabolic line and the elliptic boundary. This method is shown to be second order accurate and easy to implement. Once the elliptic problem is solved, thus giving u and its derivatives along the parabolic line, the straightforward Cauchy problem is solved in the hyperbolic region. Numerical verification of these results is presented.
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