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- W345531738 abstract "This chapter provides an overview on the penalty method. It discusses some of the nonlinear initial value problems. The penalty method is used mostly for practical purposes in numerical analysis. It has been applied to initial value problems for nonlinear partial differential equations for theoretical purposes, for instance, to prove the existence of solutions in non-cylindrical regions by several authors. In these applications, the idea or the formal procedure of approximation based on the principle of the penalty method is quite simple in general, while the rigorous proof of the convergence of the approximate solutions sometimes becomes rather delicate and involved, depending on the nature of the original problem and the choice of penalizing terms. The chapter explains this situation through initial value problems in non-cylindrical regions for some simple but important equations. Study of the initial value problem for the first equation in a non-cylindrical region has, in various aspects, many things in common with the recent theory of nonlinear semi-groups of contractions or the theory of abstract evolution equations with dissipative generators. The chapter also explains that the existence of (weak) solutions of the Navier-Stokes equations in non-cylindrical regions is obtained as a result of applying the penalty method." @default.
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- W345531738 date "1971-01-01" @default.
- W345531738 modified "2023-10-01" @default.
- W345531738 title "The Penalty Method and Some Nonlinear Initial Value Problems" @default.
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- W345531738 doi "https://doi.org/10.1016/b978-0-12-775850-3.50019-4" @default.
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