An Adaptive Fictitious Domain Method for Elliptic Problems

  • Stefano Berrone
  • Andrea Bonito
  • Marco VeraniEmail author
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 12)


In the Fictitious Domain Method with Lagrange multiplier (FDM ) the physical domain is embedded into a simpler but larger domain called the fictitious domain. The partial differential equation is extended to the fictitious domain using a Lagrange multiplier to enforce the prescribed boundary conditions on the physical domain while all the other data are extended to the fictitious domain. This lead to a saddle point system coupling the Lagrange multiplier and the extended solution of the original problem. At the discrete level, the Lagrange multiplier is approximated on subdivisions of the physical boundary while the extended solution is approximated on partitions of the fictitious domain. A significant advantage of the FDM is that no conformity between these two meshes is required. However, a restrictive compatibility conditions between the mesh-sizes must be enforced to ensure that the discrete saddle point system is well-posed. In this paper, we present an adaptive fictitious domain method (AFDM ) for the solution of elliptic problems in two dimensions. The method hinges upon two modules ELLIPTIC and ENRICH which iteratively increase the resolutions of the approximation of the extended solution and the multiplier, respectively. The adaptive algorithm AFDM is convergent without any compatibility condition between the two discrete spaces. We provide numerical experiments illustrating the performances of the proposed algorithm.


Lagrange Multiplier Physical Domain Discrete Space Saddle Point Problem Extended Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work has been partially supported by the Italian MIUR through PRIN research grant 2012HBLYE4_001 “Metodologie innovative nella modellistica differenziale numerica”, by INdAM-GNCS and by the National Science Foundation grant DMS-1254618.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.MOX-Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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