Finite-element approximation of 2D elliptic optimal design

Fréchet differentiability of the elasto‐acoustic scattered field with respect to Lipschitz domains

“…The BVP is equivalent to F .Â, v  , q  / D 0, where the operator F is given by (13) and (14). Moreover, F is continuous in a neighborhood of .0, v 0 , q 0 / from X Y to Z, and its Fréchet derivatives F 0  and F 0 w exist and are continuous in a neighborhood of .0, w 0 / in X Y.…”

“…The question of the dependence of the solutions of partial differential equations, with respect to their domains, is an important mathematical problem that has been extensively studied in an abstract framework [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], as well as for many specific application problems such as acoustic problems [17][18][19][20][21][22][23], elastic problems [24,25], conductivity problems [26], and various electromagnetic problems [27][28][29][30][31][32][33][34][35][36]. However, to the best of our knowledge, the case of elasto-acoustic scattering problems has not been studied yet in spite of its applied nature and its prevalence in engineering and numerical literature (see, e.g., [37] and the references therein).…”

Finite Element Discretization in Shape Optimization Problems for the Stationary Navier-Stokes Equation

“…For any admissible Ω ∈ O, we define its discrete approximation as follows (Chenais and Zuazua [2006] or Tiba [2011] where other variants are also discussed) :…”

“…This work is concerned with the discretization and the associated convergence analysis, in the spirit of general shape optimization problems for linear elliptic systems, as discussed in Chenais and Zuazua [2006] and in Tiba [2011]. Another approximation procedure for such problems is due to Neittaamäki, Pennanen and Tiba [2009].…”

Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

“…Whether this classical engineering practice leads to convergent algorithms is unknown in many other optimal design problems, except in some other particular examples as it occurs when dealing with the optimal shape design of the domain for Dirichlet Laplacian in two space dimensions (see [10]). Note, however, that, in the later, there is no result about the convergence rate.…”