A Petrov Galerkin finite-element method for interface problems arising in sensitivity computa\-tions
A discontinuous Petrov Galerkin finite element method is proposed and analyzed to deal with the computation of sensitivity analysis of 1D interface problems. First, to motivate the difficulties arising in the numerical solution of these problems the authors start by introducing a simple example of steady state heat conduction in a thin rod with \( \Omega=[0,1]\) consisting of two materials with the interface at the spatial location \( x=q,\) \(q \in (0,1)\). It is shown that a weak formulation is necessary to capture the discontinuities of the solution and by means of a Petrov Galerkin method with suitable trial functions they are able to establish the existence of a unique solution and to prove the convergence in suitable spaces. Further, the paper presents the results of two numerical examples to show the advantage of the proposed technique over a standard finite element approach because it avoids Gibbs phenomena in the discontinuity of the sensitivity across the interface.
scientific article; zbMATH DE number 5003108