Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

Question

1. What have the authors contributed in "Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface" ?

2. What future works have the authors mentioned in the paper "Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface" ?

3. What are the other methods accounting for complex interfaces?

4. What other techniques are used to solve the elliptic interface problem?

Accepted Answer

In this paper the authors propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients ( with general non homogeneous jumps in the solution and its gradient ) in 2D and 3D.

Accepted Answer

Extensions of the method to the case of adaptive Cartesian grids and High Performance Computing is subject of future research.

Accepted Answer

Other methods accounting for complex interfaces are the arbitrary Lagrangian Eulerian method (ALE) [23, 18], the penalization methods [55, 5, 10] and the class of Immersed Finite Volume Methods (IFVM) [21].

Accepted Answer

Sharp-edge interface problems are also solved by other techniques such as the matched interface and boundary (MIB) method [65], the finite volume method [48], and the non-symmetric positive definite finite element method for matrix coefficient [32].

Accepted Answer

For complex moving interfaces a suitable re-meshing procedurestill encumber the computational cost, and therefore the problems of this class are most efficiently approached by methods that embed the interface in a fixed Cartesian grid.

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Variable high-jump coefficients across the interface have been presented in recent methods, where only numerical tests with jump ratio up to the order of one thousand have been explored [38, 65, 47, 48], although they show some unsolved robustness issues (lack of monotonicity of the error when the grid is refined [38] or accuracy lower than second order [65]).

Accepted Answer

The tolerance for the stopping criterion of the multigrid is small enough in order to make the error of the iterative scheme negligible with respect to the one associated with the discretization error.

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The method is second-order accurate for the Poisson and heat equation, while it downgrades to first-order accuracy for the Stefan-type problem.

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in order to have a sufficient number of tests to compute the accuracy order, the authors choose the number of grid points (N + 1)3 by the following formula:N + 1 = d25× 1.4je, j = 0, . . . , 5,where d·e is the ceiling function.

Accepted Answer

Iterations are performed until a suitable tolerance is satisfied, chosen in such a way the algebraic error (due to this stopping criterion) is negligible with respect to the discretization error.

Accepted Answer

Let ∂Ω+h be the set of the ghost points for Ω +, namely the grid points outside Ω+ and such that one of the four neighbour grid points is inside Ω+, i.e.:(x, y) ∈ ∂Ω+h ⇐⇒ (x, y) ∈ Dh\\Ω + h and {(x± h, y), (x, y ± h)} ∩ Ω + h 6= ∅.

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

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Most frequently asked questions

1. What have the authors contributed in "Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface" ?

2. What future works have the authors mentioned in the paper "Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface" ?

3. What are the other methods accounting for complex interfaces?

4. What other techniques are used to solve the elliptic interface problem?

5. What is the way to solve the problem of moving interfaces?

6. How many high-jump coefficients have been presented in recent methods?

7. What is the tolerance for the stopping criterion of the multigrid?

8. What is the first-order accuracy of the ghost-fluid method?

9. How many grid points do the authors need to compute the accuracy order?

10. How many iterations are performed on the grid?

11. What is the set of the ghost points for +?

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Citations

Very high-order method on immersed curved domains for finite difference schemes with regular Cartesian grids

A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration

Lumped Element Method ? A Discrete Calculus Approach for Solving Elliptic and Parabolic PDEs

Crank–Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problems

A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

References

A finite element method for crack growth without remeshing

Level Set Methods and Dynamic Implicit Surfaces

A level set approach for computing solutions to incompressible two-phase flow

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

Level set methods and fast marching methods : evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science

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