A unified immersed finite element error analysis for one-dimensional interface problems

It has been noted that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.


Introduction
Partial differential equations with discontinuous coefficients arise in many areas of sciences and engineering such as heat transfer, acoustics, structural mechanics, and electromagnetism. The discontinuity of the coefficients results in multiple challenges in the design and the analysis of numerical methods and it is an active area of research in the communities of finite element, finite volume, as well as finite difference method.
In [7,32], Adjerid and Moon discussed IFE methods for the following acoustic interface problem where ρ, c are equal to ρ + , c + on interval (α, b) and to ρ − , c − on (a, α). Assuming that the exact solution (u, p) has sufficient regularity in (a, α) and (α, b), respectively, we can follow the idea in [31] to show that the exact solution satisfies the following so-called extended jump conditions: for certain positive constants r p k and r v k . In [7,32], IFE spaces based on polynomials of degree up to 4 were developed with these extended jump conditions, and these IFE spaces were used with a discontinuous Galerkin (DG) method to solve the above acoustic interface problem with pertinent initial and boundary conditions. Numerical examples presented in [7,32] demonstrated the optimal convergence of this DG IFE method, but we have not seen any error analysis about it in the related literature.
Extended jump conditions have also been used in the development of higher degree IFE spaces for solving other interface problems [2,4,5,13,16,28]. This motivates us to look for a unified framework for the error analysis for methods based on IFE spaces constructed with the extended jump conditions such as those in (2). As an initial effort, our focus here is on one-dimensional interface problems.
One challenge in error estimation for IFE methods is that the scaling argument commonly used in error estimation for traditional finite element methods cannot be directly applied. In the standard scaling argument, local finite element spaces on elements in a sequence of meshes are mapped to the same finite element space on the reference element via an affine transformation. However, the same affine transformation will map the local IFE spaces on interface elements in a sequence of meshes to different IFE spaces on the reference element because of the variation of interface location in the reference element, see the illustration in Figure 1. A straightforward application of the scaling argument to the analysis of the approximation capability of an IFE space will result in error bounds of a form C(α)h r , i.e., the constant factor C(α) in the derived error bounds depend on the location of the interface in the reference element, and this kind of error bounds cannot be used to show the convergence of the related IFE method unless one can show that the constant factor C(α) is bounded for allα in the reference element, which, to our best knowledge, is difficult to establish. Alternative analysis techniques such as multi-point Taylor expansions are used [5] which becomes awkward for higher degree IFE spaces, particularly so for higher degree IFE spaces in higher dimension. To circumvent this predicament of the classical scaling argument, we introduce a mapping between the related Sobolev space and the IFE space by using weighted averages of the derivatives in terms of the coefficients in the jump conditions. We show that the Sobolev norm of the error of this mapping can be bounded by the related Sobolev semi-norm. This essential property enables us to establish a Bramble-Hilbert type lemma for the IFE spaces, and, to our best knowledge, this is the first result that makes the scaling argument applicable in the error analysis of a class of IFE methods. For demonstrating the versatility of this unified error analysis framework, we apply it to establish, for the first time, the optimal approximation capability of the IFE space designed for the acoustic interface problem (1). Similarly, we apply this immersed scaling argument to the IFE space designed for an elliptic interface problem considered in [5] as well as the IFE space for the Euler-Bernoulli Beam interface problem considered in [24,26,35] leading to much simpler and elegant proofs.
The paper is organized as follows. In Section 2, we introduce the notation and spaces used in the rest of the paper. In Section 3, we restrict ourselves to study of the IFE functions on the interval [0, 1], we show that they have similar properties to polynomials, for example, they both have the same maximum number of roots, they both admit a Lagrange basis and they both satisfy an inverse and a trace inequality. In Section 4, we define the notion of uniformly bounded RIFE operators and how the scaling argument is applicable using an immersed Bramble-Hilbert lemma. In Section 5, we study the convergence of the DG-IFE method for the acoustic interface problem (1). In Section 6, we give shorter and simpler proofs for the optimal convergence of IFE methods for the second-order elliptic interface problem as well as the fourth-order Euler-Bernoulli beam interface problem.

Preliminaries
Throughout the article, we will consider a bounded open interval I = (a, b) with |a| , |b| < ∞, and let α ∈ I be the interface point dividing I into two open intervals I − = (a, α), I + = (α, b). This convention extends to any other open interval B ⊆ R with B − = B ∩ (−∞, α) and B + = B ∩ (α, ∞). For every bounded open interval B not containing α, let W m,p (B) be the usual Sobolev space on B equipped with the norm ∥·∥ m,p,B and the seminorm | · | m,p,B . We are particularly interested in the case of p = 2 corresponding to the Hilbert space H m (B) = W m,2 (B), and we will use ∥·∥ m,B and | · | m,B to denote ∥·∥ m,2,B , | · | m,2,B respectively for convenience. We will use (·, ·) B and (·, ·) w,B to denote the classical and the weighted L 2 inner product defined as Given a positive finite sequence r = (r i )m i=0 ,m ≥ 0 and an open interval B containing α, we introduce the following piecewise Sobolev space: The norms, semi-norms and the inner product that we will use on H m+1 α,r (B) are We note, by the Sobolev embedding theory, that H m+1 α,r (B) is a subspace of By dividing I into N sub-intervals, we obtain the following partition of I: We will assume that there is k 0 ∈ {1, 2, . . . , N } such that x k0−1 < α < x k0 , which is equivalent to α ∈ • I k0 . We define the discontinuous immersed finite element space W m α,r on the interval I as where P m (I k ) is the space of polynomials of degree at most m on I k and V m α,r (I k0 ) is the local immersed finite element (LIFE ) space defined as: In discussions from now on, given a function v in H m+1 α,r (or C m α,r or V m α,r ), its derivative is understood in the piecewise sense unless specified otherwise. By definition, we can readily verify that V m−1 α,r (I k0 ) ⊂ V m α,r (I k0 ) for a given finite sequence r = (r i )m i=0 ,m ≥ 1. In order to study the LIFE space V m α,r (I k0 ), we will investigate the properties of the corresponding reference IFE (RIFE ) space V m α,r (Ǐ) on the reference intervalǏ = [0, 1] with an interface pointα ∈ (0, 1). Our goal is to extend the scaling argument to such IFE spaces and use the IFE scaling argument to show IFE spaces such as V m α,r (I k0 ) have the optimal approximation capability, i.e., every function in H m+1 α,r (I k0 ) can be approximated by functions from the IFE space V m α,r (I k0 ) at the optimal convergence rate. Following the convention in the error analysis literature for finite element methods, we will often use a generic constant C in estimates whose value varies depending on the context, but this generic constant is independent of h and the interface α ∈ I k0 orα ∈Ǐ unless otherwise declared.

The dimension of
forms a basis of V m α,r (Ǐ) and will be referred to as the canonical basis.
Proof. We will prove the statements in order: which uniquely defines a polynomialφ ∓ ∈ P m (Ǐ ∓ ):
The results in Lemma 1 allows us to introduce an extension operator that mapsφ s toφ s ′ .
By Lemma 1, the extension operator E m,s ′ α,r is well-defined and is linear. Furthermore, by Lemma 1 again, this extension operator is also invertible. Consequently, the dimension of the RIFE space is the same as the dimension of the traditional polynomial space of the same degree.

Lemma 2.
There exists a constant C > 0 that depends on m such that for every wheres = ∓1 for s = ±. In particular, ifȟ s ≥ȟ s ′ , we have Proof. First, we note that (9) is a straightforward consequence of (8). Here, we only need prove (8) for s = − since the case s = + can be proven similarly. For every (φ − ,φ + ) ∈ V m α,r (Ǐ), we first defineφ − ∈ P m ([0, 1]) aŝ Now, let us writeφ + as a finite Taylor sum aroundα and useφ (i) Using (10), we can replaceφ We square and integrate (11), then we apply the change of variables z = x −α to get We can bound r i and |φ We also have m i=0 1 i! ≤ e. Using these bounds, we get Since P m ([0, 1]) is a finite dimensional space, all norms are equivalent. In particular, there is a constant C(m) such that By using a change of variables, we can show that Finally, we combine (12), (13) and (14) to get Next, we will use the bounds on the extension operator E m,š α,r to establish inverse inequalities which are independent ofα for the RIFE space. Lemma 3. Letm ≥ m ≥ 0, {r k }m k=0 ⊂ R + ,α ∈ (0, 1). Then there exists C(m, r) > 0 independent ofα such that for everyφ ∈ V m α,r (Ǐ) we have |φ| i,Ǐ ≤ C(m, r) ∥φ∥ 0,Ǐ , 0 ≤ i ≤ m + 1.
Sinceφ + = E m,+ α,r (φ − ), the formula in (17) leads to the following identity about the permutation of the classical differential operator and the extension operator: As a piecewise functionφ = (φ − ,φ + ) ∈ V m α,r (Ǐ), the value ofφ atα is not defined in general since the two sided limitsφ − (α) andφ + (α) could be different if r 0 ̸ = 1. However, ifφ s (α) = 0 thenφ s ′ (α) = 0 for s = +, −. Furthermore, the multiplicity ofα as a root ofφ − is the same as its multiplicity as a root ofφ + . This observation motivates to defineα as a root ofφ of multiplicity d ifα is a root ofφ − of multiplicity d. The following theorem shows that the number of roots of a non-zero functionφ ∈ V m α,r (Ǐ) counting multiplicities cannot exceed m (similar to a polynomial of degree m), this theorem will be crucial to establish the existence of a Lagrange-type basis in V m α,r (Ǐ) and constructing an immersed Radau projection later in Section 5.
Proof. We start from the base case m = 0 and then proceed by induction. Letφ for some c ̸ = 0. In this case,φ has no roots since N 0 α,r has no roots. Now, assume that for every positive sequence q, the number of roots of any non-zeroφ ∈ V m−1 α,q (Ǐ) is at most m − 1. Next, we will show that for a given positive sequence r, every functionφ ∈ V m α,r (Ǐ) has at most m roots by contradiction: Therefore, ξ j >α and ξ 1 ≤α becauseφ ± ∈ P m (Ǐ s ). let ξ i0 be the largest root that is not larger thanα, i.e., . It remains to show thatφ ′ has an additional root in (ξ i0 , ξ i0+1 ). To show that,we consider two cases: • ξ i0 =α: In this case,φ is continuous andφ + (α) =φ + (ξ i0+1 ) = 0. By the mean value theorem, we conclude thatφ ′ + has a root in (ξ i0 , ξ i0+1 ).
The previous theorem allows us to establish the existence of a Lagrange basis on V m α,r (Ǐ) for every choice of nodes and for every degree m which was proved by Moon in [32] for a few specific cases m = 1, 2, 3, 4. Proof.
The equations (21) form a homogeneous system of m equations with m + 1 unknowns. Therefore, it has a non-zero solution. From Theorem 1, we know thatL i (ξ i ) ̸ = 0; otherwise,L i would have m + 1 roots. This allows us to define L i as By (20), is a basis for V m α,r (Ǐ) since its dimension is m + 1 from Lemma 1.
In addition to having a Lagrange basis, the RIFE space has an orthogonal basis with respect to (·, ·) w,Ǐ as stated in the following theorem in which we also show that if a functionφ ∈ V m α,r (Ǐ) is orthogonal to V m−1 α,r (Ǐ) with respect to (·, ·) w,Ǐ , thenφ has exactly m distinct interior roots similar to the classical orthogonal polynomials. Although the theorem holds for a general weight w, we will restrict our attention to a piecewise constant function w: where w ± are positive constants. The result of this theorem can also be considered as a generalization for the theorem about the orthogonal IFE basis described in [12] for elliptic interface problems.
, and let w :Ǐ → R + be defined as in (22), then there is a non-zeroφ ∈ V m α,r (Ǐ) such that Furthermore,φ has exactly m distinct roots in the interior ofǏ.
Proof. Existence is a classical result of linear algebra. The proof of the second claim follows the same steps used for orthogonal polynomials: Note thatφ has at least one root of odd multiplicity in the interior ofǏ since (φ, N 0 α,r ) w,Ǐ = 0. Assume thatφ has j < m distinct roots {ξ i } j i=1 of odd multiplicity in the interior ofǏ. Following the ideas in the proof of Theorem 2, we can show that there isψ 0 ∈ V ǰ α,r (Ǐ) such thatψ 0 (ξ i ) = 0 for 1 ≤ i ≤ j. Furthermore, all roots ofψ 0 are simple according to Theorem 1 since the sum of multiplicities cannot exceed j, andψ 0 changes sign at these roots. This means that wφψ 0 does not change sign onǏ. As a consequence, According to Theorem 3, one can see thatφ → φ(0) 2 +φ(1) 2 defines a norm on Q m α,w,r (Ǐ) which is onedimensional. Thus, it is is equivalent to the L 2 norm and the quantity √φ depends only onα, w and r (and not on the choice ofφ ∈ Q m α,w,r (Ǐ)). Furthermore, The following lemma shows that the equivalence constant is independent of the interface location. This result will be crucial later in the analysis of Radau projections.
and w be defined as in (22) then, there exist C(m, w, r) and C(m, w, r) > 0 independent ofα such that for everyφ ∈ Q m α,w,r (Ǐ), we have Proof. The inequality on the right follows from the inverse inequality (15) for the IFE funcdtions. For a proof of the inequality on the left, see Appendix A.

An immersed Bramble-Hilbert lemma and the approximation capabilities of the LIFE space
In this section, we will develop a new version of Bramble-Hilbert lemma that applies to functions in H m+1 α,r (Ǐ) and its IFE counterpart. This lemma will serve as a fundamental tool for investigating the approximation capability of IFE spaces. In the discussions below, we will use 1 B for the indicator function of a set B ⊂ R and we define w i = r i 1Ǐ− + 1Ǐ+ for i = 0, 1, . . . , m.
. Therefore, for any given x, y ∈Ǐ, we have We integrate this identity onǏ with respect to x and use (w 0 , v (0) )Ǐ = 0 to get Taking the absolute value and applying the Cauchy-Schwarz inequality, we get where τ is the shift operator described in (17), we can use the same reasoning to show that Repeating the same arguments, we can obtain which leads to (25) with Proof.
For v ∈ H m+1 α,r (Ǐ), to see thatπ m α,r v exists and is unique, we consider the problem of findingφ ∈ V m α,r (Ǐ) such that By Lemma 1, we can expressφ in terms of the canonical basiš Then, by (28), the coefficients c ofφ are determined by the linear system Therefore, A is invertible andπ m α,r v =φ is uniquely determined by (27).
We note that the mappingπ m α,r : v ∈ H m+1 α,r (Ǐ) →π m α,r v ∈ V m α,r (Ǐ) is linear because of the linearity of integration. We now present an immersed version of the Bramble-Hilbert lemma [10] which can be considered a generalization of the one-dimensional Bramble-Hilbert lemma in the sense that if r ≡ 1, then, this immersed Bramble-Hilbert lemma recovers the classical Bramble-Hilbert lemma.
is a linear map that satisfies the following two conditions 2. There exists an integer j, 0 ≤ j ≤ m + 1 such thatP m α,r is bounded with respect to the norm ∥·∥ j,Ǐ as follows: Then, there exists C(m, r) > 0 independent ofα, such that where Proof. SinceP m α,r is a projection in the sense of (29), we haveP m α,rπ m α,r v =π m α,r v. Using the triangle inequality and Then, applying Lemma 5 and Theorem 4 to the right hand side of the above estimate leads to (31).
Next, we extend the results of Lemma 6 to the physical interface element Following the tradition in finite element analysis, for every function φ defined on the interface element I k0 , we can map it to a function Mφ =φ defined on the reference intervalǏ by the standard affine transformation: Furthermore, given a mapping P m α,r : H m+1 α,r (I k0 ) → V m α,r (I k0 ), we can use this affine transformation to introduce a mappingP m α,r : It can be verified that the mappings M, P m α,r andP m α,r satisfy the following commutative diagram: We now use the immersed Bramble-Hilbert lemma in the scaling argument to obtain estimates for the projection error v − P m α,r v.
Proof. The proof follows the same argument as for the classical case r k = 1, k = 0, 1, . . . , m [14]. We start by applying the change of variables Next, we combine Lemma 6 and (36) to obtain |v − P m α,r v| i, Nevertheless, the estimate (35) does not directly lead to the convergence of P m α,r v to v as h → 0 unless we can show that P m α,r i,j,Ǐ is uniformly bounded with respect toα ∈Ǐ, and this can be addressed by the uniform boundedness of P m α,r defined as follows.
The simplest example of a uniformly bounded collection of LIFE projections is the L 2 projectionPr m α,r : Choosing q =Pr is an uniformly bounded collection of projections. Consequently, by Theorem 6, we can obtain the following optimal approximation capability of the LIFE space. |q − v| 0,I k 0 ≤ Ch m+1 |v| m+1,I k 0 , ∀v ∈ H m+1 α,r (I k0 ).
5 Analysis of an IFE-DG method for a class of hyperbolic systems with discontinuous coefficients In this section, we employ the results of Section 3 and Section 4 to analyse an IFE-DG method for the acoustic interface problem (1). To our knowledge, the analysis of such method for hyperbolic systems has so far not been considered in the literature unlike the IFE methods for elliptic problems. The main challenge that time-dependent problems present is the plethora of possible jump coefficients. For instance, the jump coefficients r p k , r v k described in (2) for the acoustic interface problem are given by: The nature of the jump coefficients in (41) makes the study of this particular IFE space extremely tedious as observed in [32]. Fortunately, the theory developed in Section 3 and Section 4 is general and applies to any choice of positive jump coefficients.

Problem statement and preliminary results
Let I = (a, b) be a bounded interval containing α, and let ρ ± , c ± be positive constants describing the density and the sound speed in I ± , respectively. Now, we consider the acoustic interface problem on I where u = (p, u) T is the pressure-velocity couple and The matrices A ± can be decomposed as A ± = P ± Λ ± P −1 ± , where Λ ± = diag(−c ± , c ± ). Using this eigen-decomposition, we define A + ± = P ± diag(0, c ± )P −1 ± , A − ± = P ± diag(−c ± , 0)P −1 ± , and |A ± | = P ± diag(c ± , c ± )P −1 ± to be the positive part,the negative part, and the absolute value of A ± , respectively. The acoustic interface problem that we are considering here is subject to the following homogeneous inflow boundary conditions and interface condition u(α − , t) = u(α + , t), t ≥ 0.
In the remainder of this section, let S ± = diag(ρ −1 ± c −2 ± , ρ ± ) and S(x) = S ± if x ∈ I ± , then Now, we can multiply (42a) by S and write the acoustic interface problem as Lombard and Piraux [30], and Moon [32] have shown that by successively differentiating u(·, t) α = 0, where · is the jump, we obtain Since R k is diagonal (see part (a) of Lemma 8), the condition u (k) (α + , t) = R k u (k) (α − , t) is equivalent to where r p k and r u k are defined in (41). These decoupled interface conditions make the results obtained previously about the approximation capabilities of the LIFE space directly applicable to vector functions in the product spaces where r = (r p , r u ). Now, we define the following bilinear forms where the numerical fluxŵ( . Now we define the immersed DG formulation as: We note that the discrete weak form (47) and the discrete space W m α,r (T h ) are identical to the ones described in the IDPGFE formulation in [32].
Next, we will go through some basic properties of the matrices S ± and A ± , these properties will be used later in the proof the L 2 stability in Lemma 9, in the analysis of the immersed Radau projection and in the convergence estimate. (d) Let s,s ∈ {+, −}, and let w ∈ R 2 , then wheres ′ is dual ofs defined at the beginning of Section 3, and ∥·∥ is Euclidean norm.
(e) Let s ∈ {+, −}. Then, there is a constant C(ρ s , c s ) > 0 such that Proof. (a) We have A 2 ± = c 2 ± Id 2 , where Id 2 is the the 2 × 2 identity matrix. Therefore, Using (50), we immediately obtain A −2k , where r p k and r u k are defined in (41).

(b) Let
then, by a simple computation, we can show that S s = P −T s P −1 s and A s = P s diag(−c s , c s )P −1 s .
(c) We have S s A + s = P −T s diag(0, c s )P −1 s , where P s is defined in (51). Therefore, S s A + s is a symmetric semipositive definite matrix. The other two claims can be proven similarly.
(d) We will only consider the cases = + here, the other case can be proven similarly. Consider a vector w ∈ R 2 that satisfies Now, letw = P s w where P s is defined in (51), then (52) can be written as Since both norms are non-negative, we have diag(−c s , 0)w = 0 and P s diag(0, c s )w = 0. P s being invertible, we getw = 0. Consequently, w = P −1 sw = 0.
(e) We have by direct computation Lemma 9. Let u be a solution to (42), and let ϵ(t) = 1 2 (u(·, t), S(·)u(·, t)) 2 0,I , then Proof. By multiplying (44) by u T and integrating on I ± , we obtain The matrices S andÃ are symmetric. Therefore, we rewrite the previous equation as Since u is continuous at α (from (42e)), we have Now, we can rewrite the term u(b, t) TÃ u(b, t) as where the last equality follows from the boundary condition (42c). Since SA + is symmetric semi-positive definite (see part (c) of Lemma 8), we conclude that u(b, t) TÃ u(b, t) ≥ 0. Similarly, we have u(a, t) TÃ u(a, t) ≤ 0. Therefore, The previous lemma shows that ϵ(t), interpreted as the energy of the system, is decreasing. This is to be expected since the boundary conditions in (42c) are dissipative (see [9]). Furthermore, if we let ϵ h (t) = 1 2 M (u h (·, t), u h (·, t)) be the discrete energy, then The proof of (54) follows the same steps as the scalar case described in [15].

The immersed Radau projection and the convergence analysis
We denote by Ru ∈ W m α,r (T h ) the global Gauss-Radau projection defined as Although Ru is a global projection, it can be constructed on each element independently, the construction of (Ru) |I k where k ̸ = k 0 can be found in [1,36] for the scalar case and can be generalized easily to systems. On the interface element, we define the local immersed Radau projection (IRP ) operator Π m α,r : H m+1 α,r (I k0 ) → V m α,r (I k0 ) using (34) as is called the reference IRP operator and it is defined as the solution to the following system: Next, we will go through some basic properties of the IRP to prove that the IRP is well defined and is uniformly bounded on the RIFE space V m α,r (Ǐ). From there, we can show that the IRP error on the LIFE space V m α,r (I k0 ) decays at an optimal rate of O(h m+1 ) under mesh refinement.
To finalize the proof, we only need to show that (56) can be written as a square system. Let A ± = P ± diag(−c ± , c ± )P −1 ± be an eigen-decomposition of A ± . Then, (56) can be written as which is a system of 2(m + 1) equations with 2(m + 1) variables. Since the homogeneous system admits at most one solution, we conclude that (56) has exactly one solution.
By combining Lemma 11 and Lemma 12, we can show that the norm ofΠ m α,rǔ can be bounded by a norm ofǔ independently ofα as described in the following theorem. We note thatΠ m α,r maps H m+1 α,r (Ǐ) to V m α,r (Ǐ). Nevertheless, we shall callΠ m α,r a RIFE projection since the results from Section 4 in the scalar case apply directly to the vector case here.

Novel proofs for results already established in the literature
In this section, for demonstrating the versatility of the immersed scaling argument established in Section 3 and Section 4, we redo the error estimation for two IFE methods in the literature. One of them is the IFE space for an elliptic interface problem [5], and the other one is the IFE space for an interface problem of the Euler-Bernoulli Beam [24]. We note that the approximation capability for these IFE spaces were already analyzed, but with complex and lengthy procedures. Our discussions here is to demonstrate that similar error bounds for the optimal approximation capability of these different types of IFE spaces can be readily derived by the unified immersed scaling argument.
6.1 The m-th degree IFE space for an elliptic interface problem In this subsection, we consider the m-th degree IFE space developed in [5] for solving the following interface problem: Assume that f is in C m−1 (I) which implies that the solution u ∈ H m+1 α,r (I) with The discussion in Section 2 suggests the following IFE space for this elliptic interface problem: which coincides with the one developed in [5] based on the extended jump conditions where it was proved, by an elementary but complicated multi-point Taylor expansion technique, to have the optimal approximation capability with respect to m-th degree polynomials employed in this IFE space. We now reanalyze this IFE space by the immersed scaling argument. The continuity of functions in the IFE space suggests to consider the following immersed Lobatto projection L m α,r : H m+1 α,r (I k0 ) → V m α,r (I k0 ) defined by where τ 2 = τ • τ and τ is the shift operator defined in (17). The related reference immersed Lobatto projectioň L m α,r : H m+1 α,r (Ǐ) → V m α,r (Ǐ) is defined by the diagram (34), that is,Ľ m α,rǔ = L m α,r u whereǔ = Mv. For simplicity, letũ =Ľ m α,rǔ for a givenǔ ∈ V m α,r (Ǐ) and note that the system (87) is a square system of m + 1 equations since the last line can be written as m − 1 equations. Therefore, we only need to show that ifǔ ≡ 0 theñ u ≡ 0 to prove thatĽ m α,r is well defined.
Lemma 13. The reference immersed Lobatto projectionĽ m α,r is well defined.
Next, we will show that {Ľ m α,r } 0≤α<1 is a uniformly bounded collection of RIFE projections in the following lemma.
Then, we can use Theorem 6 to derive an error bound for the Lobatto projection L m α,r u in the following theorem which confirms the optimal approximation capability of the IFE space established in [5] by a more complex analysis.
Proof. This follows immediately from Lemma 14 and Theorem 6.

Euler-Bernoulli Beam interface problem
In this subsection, we apply the immersed scaling argument to reanalyze the cubic IFE space developed in [26] and [35] for solving the following interface problem of the Euler-Bernoulli beam equation: where the solution u satisfies the following jump conditions at α First, let r = 1, 1, β − β + , β − β + be fixed throughout this subsection. Then, the usual weak form of (89) suggests to consider the following IFE method: where Q 3 α,r (T h ) = H 2 0 (I) ∩ W 3 α,r (T h ). We note that the IFE space Q 3 α,r (T h ) as well as the method described by (90) were discussed in [26] and [35], and an error analysis based on a multipoint Taylor expansion was carried out to establish the optimality of this IFE method in [24]. As another demonstration of the versatility of the immerse scaling argument, we now present an alternative analysis for the optimal approximation capability of this IFE space. This new analysis based on the framework developed in Section 3 and Section 4 is shorter and cleaner than the one in the literature.
As usual, for the discussion of the approximation capability of the IFE space, we consider the interpolation on the reference elementǏ and map it to the physical element I k0 . To define the interpolation, we let {σ i } 4 i=1 be the Hermite degrees of freedom, that is, It is known [26,35] that there is a basis {L iα ,r } 3 i=0 of V 3 α,r (Ǐ) that satisfies These basis functions can then be used to define an immersed Hermite projection/interpolation operatorŠα ,r : Lemma 15. Let β ± > 0 andα ∈ (0, 1), then Proof. See Appendix B Now, we are ready to establish that {Šα ,r } 0<α<1 is a collection of uniformly bounded of RIFE projections.
Theorem 11. Let β ± > 0, i ∈ {0, 1, 2, 3}, α ∈ I k0 . Then, there is a constant C independent of α such that the following estimate holds for every u ∈ H 4 α,r (I k0 ) This theorem establishes the optimal approximation capability of the IFE space Q 3 α,r (T h ) which was first derived in [24] with a lengthy and complex procedure.

Conclusion
In this manuscript we developed a framework for analyzing the approximation properties of one-dimensional IFE spaces using the scaling argument. We have applied this IFE scaling argument to establish the optimal convergence of IFE spaces constructed for solving the acoustic interface problem, the elliptic interface problem and the Euler-Bernoulli beam interface problem, respectively. We are currently working on extending these results to IFE spaces and methods for solving interface problems in two and three dimensions.
For simplicity, let q i ∈ P m ([0, 1]) be the monomial basis q i (x) = x i for 0 ≤ i ≤ m. Using the equivalence of norms, one can show that there is c 1 (m) > 0 such that Unfortunately, if we extend (94) to V m α,r , then the constant on the right might depend onα and might grow unboundedly asα → 0 + or asα → 1 − . To circumvent this issue, we will use a scaling trick similar to the one used in the proof of Lemma 2. First, we bound (φ, q i ) ws,Ǐ s as shown in the following lemma Proof. Sinceφ ∈ Q m α,w,r (Ǐ), we have Then, by Cauchy-Schwarz inequality and (9), we have The previous lemma shows that (φ, q i )Ǐs will approach 0 if h s approaches 1. This will allow us to obtain a restricted version of (94).
for some rational functions R i,m w,r m i=0 ofα. Proof. We will prove Lemma 19 via strong induction. First, the case m = 0 is obvious. Now,we assume that For every i = 0, 1, . . . , m − 1, there are rational functions R j,i w,r ofα such that O ǐ α,w,r = i j=0 R j,i w,r (α)N ǰ α,r .
From the strong induction assumption and (100), we conclude that R j,m w,r is a rational function.
By letting C(m, w, r) = min(C 1 (m, r), C 2 (m, w, r)), we know that O m α,w,r satisfies inequality (24) stated in Lemma 4. Consequently, the estimates in (24) of Lemma 4 is true for every function in Q m α,w,r (Ǐ) because it is a one-dimensional space, and Lemma 4 is proven.