**Title:** Numerical Solution of Two Free Boundary Problems

**Speaker:** Yongmin Zhang

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

Many free boundary problems for partial differential equations can be formulated as variational inequalities. Such formulations are attractive from the point of view of numerical approximation, since the free boundary does not need to be represented directly. The basic idea is that the free boundary problem is converted to a minimization problem, which is them approximately solved. If one wants to minimize a nonlinear functional it is often fruitful to consider the relationships which must hold at a minimum. If the functional is differentiable and the permitted variations at the minimum constitute a linear space this process gives equations that the minimum must satisfy, and if the functional is quadratic these equations are linear. However, if the set of permitted variations is constrained (for example to nonnegative functions) or the functional is nondifferentiable, then one may find inequalities instead of equations. We are interested in numerically approximating solutions of two types of variational inequalities. The first one is variational inequalities with constrained admissible set, frequently called obstacle problems. The second type is variational inequalities with a non-differentiable term. An important example of this type is rigid visco-plastic Bingham fluid. $L^{\infty}$-error estimates for numerical solutions of obstacle problems have been investigated by C. Baiocchi and J. Nitsche. Though Nitsche's estimate is optimal($O(h^2|lnh|)$), the discrete solution he defined is not in general computable because the obstacle itself is not discretized. A new monotonicity principle for a discrete obstacle problems is applied to obtain an optimal $L^{\infty}$-error estimate for an approximation in which the obstacle is only respected at the vertices of the triangulation. This result both uses and improves the Nitsche's estimate. Numerical computation of Bingham fluid flow has been studied by M. Fortin and R. Glowinski, but error estimates are not yet available for their methods. A new numerical method for approximate solution of time-dependent flow of Bingham fluid in cylindrical pipes which uses regularization of nondifferentiable term is studied. Error estimates are described for the case in which the discretization is done using piecewise linear finite elements in space and backward differencing in time.

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