A relaxation procedure for domain decomposition methods using finite elements
Numerische Mathematik
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A PDE sensitivity equation method for optimal aerodynamic design
Journal of Computational Physics
The immersed finite volume element methods for the elliptic interface problems
Mathematics and Computers in Simulation - Special issue from IMACS sponsored conference: “Modelling '98”
A note on the use of transformations in sensitivity computations for elliptic systems
Mathematical and Computer Modelling: An International Journal
Differentiability with respect to parameters of weak solutions of linear parabolic equations
Mathematical and Computer Modelling: An International Journal
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Continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design, to fast algorithms in computational fluid dynamics to the quantification of uncertainty. In order to make use of these methods for interface problems, one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces. In this paper we develop a discontinuous Petrov Galerkin finite-element scheme for solving the sensitivity equation resulting from a 1D interface problem. The 1D example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities. In particular, the sensitivity boundary value problem must be formulated in a very weak sense, and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes. Numerical examples are presented to illustrate the benefits of this approach.