Q2. What are the future works mentioned in the paper "Matched interface and boundary (mib) method for elliptic problems with sharp-edged interfaces" ?
The present work provides a solution to this class of problems on the Cartesian grid by extending the marched interface and boundary ( MIB ) method [ 60,61 ] previously designed for straight or curved interfaces. Second, the standard ( higherorder ) central finite difference ( FD ) schemes are utilized so that the condition number of the discretization matrix is relatively low, and efficient linear algebraic equation solvers can be used.
Q3. What is the concept of secondary fictitious values?
The concept of secondary fictitious values is essential for overcoming the geometric difficulties created by sharp-edged interfaces.
Q4. What is the challenging problem in the field?
One of the most challenging problems in the field is the solution of elliptic equations with sharp-edged coefficients, i.e., non-smooth interfaces.
Q5. What is the main feature of the MIB?
One feature of the MIB is that it disassociates between the discretization of the elliptic equation and the enforcement of interface jump conditions.
Q6. What is the nature of the solution of irregular interfaces?
For irregular interfaces, it is nature to construct a solution in the finite element method formulation [2,9,38], in particular, using the discontinuous Galerkin technique [22].
Q7. What is the effect of local mesh refinement on the solution of irregular interfaces?
dramatic local mesh refinement is required in the vicinity of sharp edges [13], and leads to severe increase in computational time and memory requirement.
Q8. What is the way to handle a C1 irregular interface?
The standard MIB method can always handle a C1 irregular interface even if it has large curvature, since enough auxiliary points can always be found by refining mesh.
Q9. How can the present method be used to solve smooth interfaces?
For the problems with smooth (say C1) interfaces, the present method can easily achieve 2nd order convergence since required auxiliary points can always be found by refining the mesh.
Q10. How do the authors treat (bux)x and (buy)y?
To restore the order of convergence of the discretization at irregular points, the authors need to treat (bux)x and (buy)y to the designed order of convergence near the interface.
Q11. What is the function of the expansion of a fictitious value?
If only one of F component, Fi (i 2 {1,2,3,4}), is the fictitious value whose expansion is to be determined and the other three are function values or have expansions in term of function values, then the jump condition of u(x,y) is enough for the determination of Fi:uþ u ¼ ½u : ð24ÞLet wþ0 ¼ ðwþ0;i 1;wþ0;i;wþ0;iþ1Þ and w 0 ¼ ðw 0;i;w 0;iþ1;w 0;iþ2Þ.
Q12. What is the purpose of the present work?
In the present work, the authors further generalize the MIB method to allow the presence of sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with the domain boundary.
Q13. What is the main difference between the MIB approach and the FD method?
The MIB approach makes use of fictitious domains so that the standard high order central finite difference (FD) method can be applied across the interface without the loss of accuracy.
Q14. What is the default method for determining the fictitious value of a point?
If (three surrounding points are on the same side or have secondary fictitious value on the same side), then * call On-interface scheme 3 – else pause If ((i, j) is an irregular point and (i, j) 2 X ), use primary fictitious values and call the standard five-point FD scheme.
Q15. What is the problem with the previous scheme?
This difficulty is not resolvable by the previous MIB scheme because the auxiliary points in the ‘improper’ region leads to inconsistence between the number of unknowns and the number of jump conditions.