Mixed finite element methods for quasilinear second order elliptic problems: The \(p\)-version. (English) Zbl 0783.65076

The authors analyzed in two previous separate papers (a) the \(p\)-version of the finite element method for a linear second order elliptic problem [the second author, Math. Comput. 54, No. 189, 1-19 (1990; Zbl 0687.65101)]; (b) the \(h\)-version of a mixed finite element method for a quasilinear second order elliptic problem [the first author, Math. Comp. 44, 303-320 (1985; Zbl 0567.65079]).
In the present paper they combine the investigations of the two above articles to get new results for the \(p\)-version of quasilinear second order elliptic problems in a mixed weak form. They derive error estimates in the \(L^ 2\)-norm not only for the solution \(u\) itself but also for its flux. In comparison with their results with the \(h\)-version they suppose that the interpolation error can be improved.
Another point of interest for further investigations is the question whether the \(p\)-version has the double convergence rate in the presence of singularities, a well-known observation in finite element theory.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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