Abstract:
A mixed finite element method for solving the Dirichlet problem for
a fourth-order quasilinear elliptic equation in divergent form was
proposed and investigated. It was assumed that the domain in which
the problem is solved is bounded and has a dimension greater or
equal to two. When constructing the finite element scheme, all the
second derivatives of the required solution were chosen as auxiliary
unknowns. The usual triangulation of the domain by Lagrangian
simplicial (triangular) elements of orders two and higher was used.
Under the assumption that the operator of the original problem
satisfies the standard conditions of bounded nonlinearity and
coercivity, the existence of an approximate
solution for any value of the discretization parameter was proved. The uniqueness of the
approximate solution was established under tighter restrictions, namely, assuming
the Lipschitz-continuity and the strong monotony of the differential operator.
Under the same conditions, a two-layer iterative process was constructed,
and the estimation of the convergence rate independent of the discretization parameter
was proved. Accuracy estimates for the approximate solutions, optimal in the case
of linearity of the differential equation, were obtained. The results of the application
of the proposed technique to the problem of the equilibrium of a thin elastic plate
were presented.
Keywords:mixed finite element method, accuracy estimates, iterative method, convergence rate estimates, theory of plates.
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