Estimates of constants in error estimates for H^2 conforming finite elements for regularized nonlinear elliptic geometric evolution equations and question of optimality
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Geometric evolution equations in level set form are usually singular and a well-known regularization procedure generates a family of approximating non-singular equations (e.g. useful for analytical or numerical aspects). In the previous work [13] upper bounds for constants which appear in the standard finite element error estimates for elliptic regularized geometric evolution equations in dependence on the regularization parameter have been addressed and an exponential relation in the inverse regularization parameter has been observed. In this paper the aim is twofold: First, we extend the results from [13] to H^2(Ω) conforming approaches which are of interest in the special case of higher regularity of the solution in order to detect level sets, and second, we present a strong indication that the previously mentioned exponential bound which carries over to the higher conforming case is optimal (independent from the degree of conformity). This is in accordance with practical experience in own previous work and suggests at least in the special case of mean curvature flow that a parabolic approach is preferable (in order to get better FE error estimates).
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