Abstract
It is well known that a wide class of obstacle and unilateral problems arising in pure and applied sciences can be studied in a general and unifield framework of variational inequalities. In this paper, we derive the error estimates for the finite element approximate solution for a class of highly nonlinear variational inequalities encountered in the field of elasticity and glaciology in terms ofW1,p(Ω)andLp(Ω)-norms. As a special case, we obtain the well-known error estimates for the corresponding linear obstacle problem and nonlinear problems.
Highlights
Variational inequality theory is an interesting branch of applicable mathematics, which provides us with a uniform framework for studying a large number of problems occurring in different branches of pure and applied sciences, and gives us powerful and new numerical methods of solving them
The finite element techniques are being applied tb compute the approximate solutions of various classes of variational inequalities
Relative to the linear variational inequalities, little is known about the accuracy and convergance properties of finite element approximation of nonlinear variational inequalities associated with nonlinear elliptic boundary value problems
Summary
Variational inequality theory is an interesting branch of applicable mathematics, which provides us with a uniform framework for studying a large number of problems occurring in different branches of pure and applied sciences, and gives us powerful and new numerical methods of solving them. Our results represent a substantial generalization and improvement of the error analysis of finite element approximation of strongly nonlinear monotone operators and variational inequalities contributed by Glowinski and Marroco [1], Oden and Reddy [10] and Noor [9]. For piecewise linear dements and result obtained by Oden and Reddy [10] under the assumption that all the solutions lie in the interior of the closed convex set K in wl’P-space. In this way, our results represent an improvement of their result. For < p _< 2, our results appear to be new ones and there is no counterpart in the linear theory
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