New Cartesian grid methods for interface problems using the finite element formulation

TLDR: New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients, and these new methods can be used as finite difference methods.

Abstract: New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis.

Figures

Fig. 3.7. Errors of finite element solutions obtained from the conforming basis function in the maximum norm versus the mesh size h in log− log scale with r0 = π/6.28. (a): β− = 1, β+ = 1000. The linear regression analysis gives ||u−uh||∞ ≈ 6.85126h2.01002; (b): β− = 1000, β− = 1. The linear regression analysis gives ||u−uh||∞ ≈ 5.65703h2.01542.

Table 3.1 Comparisons of errors of the FEMCGAN and the IFE methods, where e0(h) and ea(h) are errors of a numerical solution in the L2(Ω) and energy norms respectively. The example is the same as the example in Section 2.5 for the case when β− = 1, β+ = 1000.

Fig. 2.3. The errors of the finite element solutions obtained in the non-conforming IFE space in the maximum norm versus the mesh size h in log− log scale with r0 = π/6.28, α = 3. (a): β− = 1, β+ = 1000. The linear regression analysis gives ||u− uh||∞ ≈ 0.64657h1.56459; (b): β− = 1000, β+ = 1. The linear regression analysis gives ||u− uh||∞ ≈ 2.79434h1.94833.

Fig. 3.4. A sub-triangle contains none of the angles of the master triangle, and the interface intersects one of the right legs and the hypotenuse of the master triangle.

Fig. 1.1. A rectangular domain Ω = Ω+ ∪Ω− with an immersed interface Γ. The coefficients β(x) may have a jump across the interface.

Fig. 2.1. A typical triangle element with an interface cutting through. The curve between D and E is part of the interface curve Γ which is approximated by the line segment DE. In this picture, T is the triangle 4ABC, T+ = 4ADE, T− = T − T+, and Tr is the region enclosed by the DE and the arc DME.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "New cartesian grid methods for interface problems using the finite element formulation" ?

In this paper, Cartesian triangulations are used for finite element methods for elliptic interface problems with discontinuities in the coefficients. 

Q2. What is the solution of u(x, y) in the conforming IFE?

if the source term f(x, y) ∈ L2(Ω) is also γth-Hölder piecewise continuous for γ > 0, then the solution u(x, y) is piecewise twice differentiable, see [12]. 

Q3. What is the function uI(x) that can approximate u(x)?

Given a function u(x) which is continuous on the entire domain and satisfies the flux jump condition, the authors define its interpolant in the IFE space Sh(Ω) as the function uI(x) ∈ Sh(Ω) such thatuI(x) = u(x), if x is a node of Th.The authors would like to know how well uI(x) can approximate u(x). 

Q4. What is the procedure to construct a local basis function in an interface element?

The authors use the standard five dimensional Euclidean vector ei (whose i-th entry is unity while the other entries are zero) to assign values of a local basis function ψi(x, y) at the vertices A,B,C, F and I, and this basis function is piecewisely constructed as follows: 

Q5. What is the linear regression analysis of Fig. 3.7?

The linear regression analysis shows that data in Fig. 3.7 obey||u− uh||∞ ≈ 6.85126h2.01002, ρ = 1 : 1000,||u− uh||∞ ≈ 5.65703h2.01542, ρ = 1000 : 1,which suggest that the conforming IFE finite element solution has a second order convergence rate in the maximum norm. 

Q6. What is the boundary condition and the source term fc?

The boundary condition and the source term fc are determined from the exact solutionu(x, y) = rα β− , if r ≤ r0, rαβ+ +( 1β− − 1 β+) rα0 , otherwise, (2.40) where r = √x2 + y2 and α = 3. 

Q7. What is the procedure to construct the non-conforming local basis functions in a typical interface?

P1. Use the values at the vertices A, B, C, F and The authorto form the three non-conforming IFE functions defined on the elements ∆ABC, ∆AFB and ∆ACI respectively. 

Q8. What is the normal direction of the line segment DE?

The authors assume that the coordinates at A, B, C, D, and E are(0, h), (0, 0), (h, 0), (0, y1), (h− y2, y2),(2.2)with the restriction0 ≤ y1 ≤ h, 0 ≤ y2 < h.(2.3)Once the values at vertices A, B, and C of the element T are specified, the authors construct the following piecewise linear function:u(x) = { u+(x) = a0 + a1x + a2(y − h), if x = (x, y) ∈ T+, u−(x) = b0 + b1x + b2y, if x = (x, y) ∈ T−, (2.4)u+(D) = u−(D), u+(E) = u−(E), β+ ∂u∂n+= β− ∂u∂n− ,(2.5)where n is the unit normal direction of the line segment DE. 

Q9. What is the reason why the authors believe that piecewise smooth requirement of the solution can be relaxed?

The authors believe that piecewise smooth requirement of the solution in the theorem can be relaxed to piecewise H2(Ωi) by developing corresponding interpolation theory in the Sobolev space. 

Q10. What is the simplest way to determine the linear function u(x)?

Th as follows:Sh(T ) = { {u(x) | u(x) is linear on T} , if T is a non-interface element, {u(x) | u(x) is defined by (2.4)-(2.5)} , if T is an interface element. 

Q11. What is the definition of the local space of shape functions for an interface element?

Remark 3.1. Actually Sh(T ), the local space of shape functions for an interface element is just the five dimensional space of continuous piecewise linear functions on the three subtriangles. 

Q12. What is the simplest explanation for the problem of u in the conforming IFE space?

2Remark 3.3.• The finite element solution u(x) in the conforming IFE space belongs to H1(Ω) but generally is not in H2(Ω) if β(x, y) has a discontinuity across the interface, see Fig. 3.5. 

Q13. What is the union of the mis-matched region of the line segments and the interface?

∑T\\Tr is the union of the mis-matched region of the line segments and the interface as shown in Fig. 2.1. From Theorem 3.2, the authors know that u− uI and its first derivatives are of O(h2) and O(h), respectively, in the maximum norm on T\\Tr of an element T , therefore, u − uI should be of O(h) in the H1 norm on the unions of these regions as well.