Convergence for an Immersed Finite Volume Method for Elliptic and Parabolic Interface Problems

In this article we analyze an immersed interface ﬁnite volume method for second order elliptic and parabolic interface problems. We show the optimal convergence of the elliptic interface problem in L 2 and energy norms. For the parabolic interface problem, we prove the optimal order in L 2 and energy norms for piecewise constant and variable di ﬀ usion coe ﬃ cients respectively. Furthermore, for the elliptic interface problem, we demonstrate super convergence at element nodes when the di ﬀ usion coe ﬃ cient is a piecewise constant. Numerical examples are also provided to conﬁrm the optimal error estimates.


Introduction
Elliptic and parabolic interface problems with discontinuous coefficients appear in a variety of disciplines, such as electromagnetism, fluid dynamics and material science. These problems can be solved by standard finite element methods (FEM) using carefully tailored meshes to resolve the interface. If the grid lines of the fitted finite element mesh are not conformed with the interface, the solution has low global regularity due to the discontinuity of the coefficients across the interface. However, high quality mesh generation is difficult and has more computational cost for some complicated geometries and interfaces. To overcome the limitations of standard fitted mesh methods many numerical methods have been developed in the past several decades.
One of the more commonly used such a method is the immersed finite element (IFE) method (Li, Z., 1998) based on Cartesian meshes. The method uses special basis functions for the interface elements while allowing the interface to immerse in the regular elements. The local interface basis functions are designed to satisfy the jump conditions at the interface while their meshes do not have to be conformed with the interface (Li, Z., 2011;Li, Z., Lin, T., Wu, X., 2003;Li, Z., Ito, K., 2006). The solution of the one dimensional interface problem is second order accurate in the infinity norm (Li, Z., 2011). Li, et al. (Li, Z., Lin, T., Rogers, R.C, 2004) analyzed the second order elliptic interface problem and their results showed IFE has approximating capability similar to that of standard FEM based on body fitting partitions. The Convergence of the IFE method for semi-linear parabolic interface problem was analyzed by Attanayake, et al. (Attanayake, C, Senaratne, D., 2011) and their results prove that the convergence of the semi-discrete solution was of the optimal order in L 2 and energy norms and the fully discrete scheme based on the backward Euler method has optimal order in L 2 norm. However, some numerical results demonstrate that IFE methods have larger point-wise error over the interface elements.
Finite volume method (FVM) is another numerical method based on Cartesian meshes. Since the method inherits local conservation of physical quantities such as, mass, momentum, and flux, FVM is used in solving problems in science and engineering (Cai, Z., 1991;Lin, Y., Liu, J., Yand, M., 2013). Ewing and his colleagues (Ewing, R., Li, Z., Lin, T., Lin, Y., 1991) are the first to investigate immersed finite volume methods (IFVM) on second order elliptic interface problems and they obtained optimal error estimates in the energy norm. Cao and his colleagues (Cao, W., Zhang, X., Zhang, Z., Zou, Q., 2017) studied the convergence of the one dimensional elliptic problem for any order finite volume schemes and produced some super convergence properties as well. A second order convergence in L ∞ norm is obtained for an elliptic interface problem using IFVM in (Wang, Q., Zhang, Z., Wang, L., 2021). However, theoretical analysis of immersed finite volume methods on parabolic interface problems remains sparse in the literature. and obtain optimal error estimates for the approximated solution. More specifically, we use so called covolume method. This method uses two types of meshes, the primal and the dual partition, associated with the trial and test spaces respectively. The exact solution of the equation is approximated in the primal partition, while the equation is discretized in the dual partition. With a transfer operator from trial space to test space, we deduce the relation to the Galerkin finite element method. As a result, we can use the properties of the finite element analysis to estimate optimal convergence rates. This is one of the main advantages of the covolume method. The theoretical framework of the covolume method was developed by Chou and his colleagues in the articles (Chou, S.H., Li, Q., 1999;Chou, S.H., Ye, X., 2007) and references therein. We extend these convergence properties to the parabolic problem. To the best of the author's knowledge, this is the first study that demonstrates convergence properties of the parabolic interface problem via the immersed finite volume method.
The rest of the paper is organized as follows. In Section 2, we introduce the immersed interface finite volume method and the convergence analysis for the elliptic interface problem. In particular, we show super convergence at the element nodes when the diffusion coefficient is a piecewise constant. The convergence for the parabolic interface problem is discussed in Section 3. Simulation of elliptic and parabolic interface problems are provided in Section 4 to confirm the theory. The conclusions are given in Section 5.

Elliptic Interface Problem
In this section we define an immersed interface finite volume method to solve a second order elliptic interface problem with the jump conditions on the interface α ∈ (0, 1) where Here, the diffusion coefficient β has a finite jump across the interface. First, consider the primal partition independent of the interface as Denote h i = x i+1 − x i and h = max 0≤i<N h i . We call the element [x k , x k+1 ] an interface element which contains α, and the other elements [x i , x i+1 ] i k are noninterface elements. For the corresponding trial space, basis functions on noninterface elements are defined as standard linear Lagrange nodal basis functions φ i , i k. On interface elements basis functions φ k and φ k+1 are defined enforcing the jump conditions The basis functions for the interface elements are given by where Vol. 15, No. 2;2023 So the trial space is chosen as,

Journal of Mathematics Research
In this space define the inner-product and the broken semi-norm and the norm for the above space V h as To construct the dual partition we choose the midpoints between nodes in the primal partition. We denote and the dual partition is given by The corresponding test space consists of the piecewise constant functions with respect to the dual partition where χ x i is the characteristic function of the dual element [x i−1/2 , x i+1/2 ] associated with the primal node x i . Moreover, the convergence of the approximated solution depends on the interpolation of the solution u given by where In (Chou, S.H., 2012), Chou proved that, u I (x) has the approximation property Here u I is the usual Lagrange interpolation and u 2,α = u H 2 (0,α) + u H 2 (α,1) .
We used Sobolev embedding theorem in the last step of the proof. Similarly Now with (7), (8) and (9), and using the fact that Π * h u = Π * h u I we have, By brute force calculation on noninterface elements, since u I is linear Now we present our immersed finite volume scheme. Integrating (1) over each control volume Note that u(x 0 ) = u(x N ) = 0. We can define the finite volume bilinear form for any u and v in V h as By writing the sum over the dual partition as a sum over the primal partition, the bilinear form (10) becomes Since Π * h v is a piecewise constant in the dual partition, the sum in the equation (11) becomes Then integration by parts of N−1 i=0 x i+1 x i (βu ) vdx brings the finite volume bilinear form to Furthermore, Next we'll show the coercivity. From (11) for noninterface elements [ Similarly, using the fact that u is only piecewise linear, on the interface element [x k , x k+1 ], Combining (14) and (15) and applying them on (11) we can see that β(x k+1/2 )u + u − (x k − α) is the only possible nonepositive term. Therefore, for the sufficiently small h, particularly on the interface element there is a constant C depend on β such that a h (u, u) ≥ C(β) u 2 1,h .
Remark 1.2. Suppose that the diffusion coefficient β is piecewise constant such that then from (12) note that a h (u, v) = (βu , v ) h . In other words when β is piecewise constant finite volume bilinear form is same as Galerkin finite element bilinear form.

Convergence of the Elliptic Problem
The immersed finite volume method to solve (1) is to find and the true solution u satisfies Subtracting (18) from (19) we obtain Theorem 1.3. Let u and u h be solutions of (1) and (18) respectively. Then Proof. Using (20) with boundedness in (13) and coercivity in (16) for any v ∈ V h we have Then using the triangular inequality and the approximation property of the interpolation (7), we can conclude Theorem 1.4. Let u and u h be solutions of (1) and (18) respectively. Then Proof. We prove this using the duality argument. Let w ∈ H 2 ∩ H 1 0 (0, 1) be the solution to dual problem −(βw ) = u − u h in (0, 1) and from (7) w − w I 1,h ≤ ch w 2,α where w I ∈ V h is the usual linear interpolant of w. From 21 we obtain At the same time, (12) implies that Subtracting (25) from (24), To estimate the first term in the right we use Theorem 1.3, (23) and (22), Due to the linearity of w I , w I − Π * h w I ≤ Ch. Then the last two terms in (26) can be estimated as follows. Applying Lemma 1.1 we have Now using (22) in (27) and applying (27) and (28) in (26) we prove the theorem.
The following theorem indicates super convergence at the nodal points. To prove it we use the properties of Galerkin finite elements.
Theorem 1.5. Assume that the diffusion coefficient β is piecewise constant as in (17). Then for all nodes x i , i = 0, . . . , N.
Proof. Fix y ∈ (0, 1) and due to Remark 1.2, let G(x, y) be the Green's function satisfying By working out the closed form of G satisfying the classical formulation we see that the Green's function G, y < α takes the form [Chou, H.S, Attanayake, A, 2017)] Now let G = G(x, x i ) and use Galerkin orthogonality property, then since G ∈ V h this proves the theorem.

Parabolic Interface Problem
In this section we consider a second order semilinear parabolic interface problem of the form with initial and boundary conditions u(·, 0) = 0 in (0, 1), with the jump conditions on the interface [u] α = 0, βu α = 0, for T > 0. The semi-discrete immersed interface finite volume problem based on above weak formulation is, find We introduce the operator R h : So by theorems (1.3) and (1.4), it follows that, We separate the error into two terms as and from Theorem (1.4) we can see that Lemma 1.6. If the mesh size h is sufficiently small, Proof. Since u is piecewise linear for each noninterface elements, we can show that, And the Simpsons rule implies However, on the interface element u is only piecewise linear. That is where E k is a quadrature error depends on the u and x k+1 − x k . Now for sufficiently small enough h we find that there is a constant C where, The convergence of the immersed finite volume method in the L2 norm is derived in the following theorem.
Theorem 1.7. Let u and u h be solutions of (1) and (18) respectively. Then, there exists a positive constant C independent of h such that Proof. Since we have the error bound for ρ, we only need to obtain the error bound for θ. Then, it follows from (32) and (33) that Thus using (37) on (38) From the same argument as in (13) and by inverse inequality Similarly, Now applying (40) and (41) on (39), with the inequality ab < a 2 + b 2 /4 for a, b, > 0 and choosing small enough to absorb θ t 2 term on the right hand into left hand, we get, After integrating both sides with respect to t and using (16), and using Gronwalls lemma we obtain Since θ(0) = 0, θ(t) 2 1,h ≤ C(T )h 2 u t 2,α and hence we obtain the desired result Theorem 1.8. Let u and u h be solutions of (1) and (18) respectively. And β is a picewise constant function as defined in (17). Then, Proof. According to the remark 1.2, the immersed finite volume bilinear form reduces to the Galerkin immersed finite element bilinear form. In (Attanayake, C., Senaratne, D., 2011), authors have proved the optimal convergence of a parabolic interface problem using immersed finite element method that has the same bilinear form. Thus the proof of this theorem follows from theorem 3.1 in (Attanayake, C., Senaratne, D., 2011).

Simulation
In this section we present two numerical examples to confirm our theory.

Conclusions
We considered an immersed interface finite volume method for second order elliptic and parabolic interface problems. By assuming the diffusion coefficient β has a finite jump across the interface and is piecewise constant,we obtained the optimal convergence in L 2 and energy norms. Further, we prove super convergence at the element nodes. We obtained optimal convergence in the L 2 norm for the parabolic interface problem with piecewise constant diffusion coefficient β. When a variable diffusion coefficient is present in the parabolic interface problem, we obtained the the optimal order in energy norm.