The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

Arthur Sarthou,Stéphane Vincent,Jean-Paul Caltagirone,Philippe Angot

doi:10.4236/ojfd.2020.103015

Arthur Sarthou, Stéphane Vincent + Show 2 more

Open Access

https://doi.org/10.4236/ojfd.2020.103015

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Abstract

A new fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized accurately. The main idea is to create auxiliary unknowns at existing grid locations which increases the degrees of freedom of the initial problem. These auxiliary unknowns allow to impose various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

Highlights

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil
The method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy
The method can be implemented in any structured grid code and can deal with immersed boundary problems too

Summary

Introduction

Such a coupling is often performed thanks to fictitious domain methods, where the computational domain does not match the physical domain. A second-order accurate discretization of the spatial operators is simple to obtain; grid generation is trivial, and there is no need to remesh the discretization grid in the case of moving or deformable boundaries Concerning this last point, fictitious domain methods can be useful even on unstructured grids: Eulerian fixed unstructured grids can fit immobile obstacles,. As particular conditions, such as jump conditions, can be required on the interface, this second class of problems is often more difficult to treat

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