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A finite difference scheme for the heat equation with mixed boundary conditions on a moving domain is presented. We use an immersed interface technique to discretize the Neumann condition and the Shortley–Weller approximation for the Dirichlet condition. Monotonicity of the discretized parabolic operator is established. Numerical results illustrate the feasibility of the approach.
The shape derivative of a functional related to a Bernoulli problem is derived without using the shape derivative of the state. The gradient information is combined with level set ideas in a steepest descent algorithm. Numerical examples show the feasibility of the approach.
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