A Fourth-Order Collocation Scheme for Two-Point Interface Boundary Value Problems

A fourth-order accurate orthogonal spline collocation scheme is formulated to approximate linear two-point boundary value problems with interface conditions. The coefficients of the differential operator may have jump discontinuities at the interface point, a nodal point of the scheme. Existence and uniqueness of the numerical solution are proved. Optimal order error estimates in the maximum norm are obtained, and a superconvergence property of the numerical solution in the maximal nodal norm is proved. Numerical results are presented confirming the theoretical estimates.


Introduction
In this paper, we propose and analyse for existence, uniqueness, and convergence an orthogonal spline collocation (OSC) scheme for the numerical solution of a linear twopoint boundary value problem (BVP) with two additional interface conditions. The variable coefficients of the differential equation may have jump discontinuities at the interface point. Existence and uniqueness of the BVP were studied in [1].
The interface boundary value problems (IBVPs) are also known as diffraction problems or BVPs with transmission (transmittal) conditions. In one dimension, they may be viewed as multipoint BVPs with piecewise continuous coefficients. IBVPs originate, for example, in heat or mass transfer in composite materials or materials with thin porous barriers, elasticity of heterogeneous materials, and population genetics [2,3]. In the case of heat transfer in layered composite materials, the temperature is discontinuous across the interface whereas the heat flux is continuous. The temperature difference is proportional to the heat flux, and the proportionality constant is known as the interfacial thermal (contact) resistance.
IBVPs can be solved numerically by, for example, homogeneous finite difference schemes [4], the shooting method [5], the immersed boundary method [6], the immersed interface method [7][8][9], or the ghost fluid method [10]. A fourthorder finite element method with Hermite elements for onedimensional interface problems is studied in [11]. By changing variables, the problem can be transformed to a regular twopoint BVP for a system of differential equations [12].
In this work, the OSC method is applied with the finite element space of piecewise Hermite cubic functions, and the interface conditions are incorporated into the scheme directly. The assumption that an interface point is a grid point restricts straight forward generalization of the proposed scheme on two-dimensional problems with a curved interface. Nevertheless, the proposed scheme is a simple modification of the OSC method for a two-point BVP; it exhibits optimal fourth-order convergence, and it is simple to implement. The interface conditions are explicitly put into the stiffness matrix, and this does not change the almost block diagonal structure of the matrix of the standard, noninterface BVP. For other numerical methods for two-point IBVPs, it might not be this easy to incorporate such general interface conditions into the stiffness matrix.

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The OSC method is a collocation scheme where collocation points are the nodes of the composite Gaussian quadrature. A survey of the OSC method for solving onedimensional BVPs is given in [13]. The numerical solution is sought in a space of smooth piecewise polynomial functions. The main advantages of the OSC method are a simple formulation, a relatively easy implementation, optimal order error estimates, and the so-called superconvergence property at the nodal points.
Schwarz and de Boor proposed the OSC method in [14], where, for an th order nonlinear two-point BVP, they proved existence and uniqueness of the numerical solution and obtained the maximum norm and the maximal nodal norm convergence estimates. In [15], the authors briefly noted (see the second paragraph on page 26) that their analysis can be extended to more general problems in which differential operators may have different orders on different subintervals of the domain with appropriate side conditions at the end points of the subintervals, that is, to IBVPs. In this work, we applied that idea to linear IBVPs with general interface conditions. We follow the methodology of [14,15], and the key elements of the analysis are existence and regularity of Green's function and approximation properties of the socalled "ssuper" projectors named in honor of S. A. Pruess, who was the first to consider such projectors (the word "ssuper" is a permutation of letters in "Pruess"). The case of one interface point is studied for simplicity of presentation. The obtained results can be generalized in a straightforward manner to problems with many interface points.
The outline of the paper is as follows. In Section 2, we formulate IBVP, introduce notation, and rewrite the problem as an integral equation. In Section 3, we introduce the OSC scheme and obtain its algebraic form for a finite element basis. In Section 4, we analyze the scheme for existence and uniqueness and obtain error estimates. In Section 5, we present two numerical examples, and, in Section 6, we give our concluding remarks.
Note that V and its derivatives can only have jump discontinuities at the interface point = , where V may be doublevalued. Let Let Denote = / and = / .
In this paper, we consider the differential equation subject to the boundary conditions and the additional general linear interface conditions Functions , , and belong to 0 , and , , , , , and are reals. For example, consider the differential equation with the discontinuous coefficients , ∈ 0 , subject to the boundary conditions (4b) and one of the following common types of interface conditions: (1)  (3) a linear radiation (a thermal contact resistance) condition: In what follows, we assume that problem (4a)-(4c) has a unique solution. Let us list some of the properties of Green's function ( , ) of the problem (see [1,16]). Green's function is continuous in the square [ , ] 2 everywhere except the lines = and = , where it may have finite discontinuities. For a fixed ∈ [ , ], satisfies the differential equation (4a) almost everywhere on ( , ) and the homogeneous boundary and the interface equations (4b) and (4c). The derivative / has, in addition, the jump discontinuity across the line = .
The lines = , = , and = divide domain [ , ] 2 into 4 triangular and 2 rectangular closed subregions. Let D be the set consisting of these 6 subregions. For a functioñ( , ) defined and sufficiently smooth in [ , ] 2 and nonnegative integers and , let For any integer ≥ 0, Green's function is in +2 on every subregion in D provided that the coefficients and are in . In particular, if , ∈ 0 , then ‖ ‖ 2,2 < +∞.
In our analysis, we make use of an integral form of IBVP (4a)-(4c). Define a nonlinear operator : 1 ( , ) → 0 as follows. For any ∈ 1 ( , ), let ∈ 1 be the unique solution of the problem IBVP (4a)-(4c) is equivalent to the following integral equation: find ∈ 0 such that Indeed, if is the solution of (10) and is the solution of (8) with = , then = + − . Let 0 be Green's function of problem (8). If 0 ∈ 0 is the piecewise linear function that satisfies conditions (4b) and (4c), then the solution of problem (8) can be written in the form Using (9) and (11), operator can be written in the form The formal derivative of acting on function is

OSC Scheme
For an integer > 0, let = { } =0 be a regular partition of the interval [ , ] such that and let the interface point = be an interior partition knot with the index . Let = [ −1 , ], ℎ = − −1 , and ℎ = max{ℎ }. Let G ℎ = { ,1 , ,2 } =1 be the sequence of the Gauss points in interval [ , ] corresponding to partition ; that is, Let be the vector space of functions defined on ⋃ =1 and continuous on every subinterval . Functions in can be double-valued at the knots of . Let For an integer ≥ 0, let , be a subspace of consisting of piecewise polynomial functions such that every ∈ , restricted on every subinterval is a polynomial of degree . Let̂ℎ = 1 [ , ] ∩ 3, be the space of piecewise Hermite cubic polynomials [17,Chapter 3], and let be a vector space of modified piecewise Hermite cubic functions that may be discontinuous at the interface knot . A piecewise cubic function V in ℎ is continuously differentiable on [ , ] and [ , ], and V can be double-valued at the interface knot . We approximate IBVP (4a)-(4c) by the following OSC scheme: find ℎ ∈ ℎ such that and ℎ satisfies the boundary conditions (4b) and the interface conditions (4c).
To obtain an algebraic form of the OSC problem, we introduce a Hermite type finite element basis in ℎ . For = 0, . . . , , define the value and the slope basis functions V and in̂ℎ, respectively, by The sequence {V , } =0 is a basis for̂ℎ. We modify this set to a basis for ℎ . Define discontinuous value basis functions and, similarly, define discontinuous slope basis functions and . The sequence is a finite element basis for ℎ . Let Let = ( ) ∈ 2 × 4 be the matrix corresponding to the interface conditions (4c), and let, for = 1, . . . , , Similarly, define matrices and +1 by replacing functions with index in and +1 by the corresponding discontinuous basis functions. Let and note that it is a block diagonal matrix whose every two consecutive 2 × 4 diagonal blocks have the two-column on top and bottom, respectively, where and are the coefficients of the boundary conditions (4b). Matrix has an almost block diagonal structure described in [18]. Note that the 2 × 4 interface condition matrix naturally "fits" into the matrix and does not change its structure. Let f = ( , 1 , . . . , −1 , 1 , 2 , +1 , . . . , , ) , be the right hand side vector. The OSC problem (18) can be written in the algebraic form The blocks of̃can be computed using the representation where = diag( ( ,1 ), ( ,2 )) and = diag( ( ,1 ), ( ,2 )), and where V ( ) = V.

Convergence Analysis
In the sequel, is a generic positive constant independent of partition . The following is the main result of this paper on existence, uniqueness, and convergence of the OSC scheme (18).
(2) If , , ∈ , ≤ 2, then (3) If , , ∈ 2+ , then, (a) for ≤ 1 and ≤ 2, where Inequalities in (30) are the maximal nodal norm error estimates. Note that the convergence is of order 4 for the derivative ℎ if ∈ 6 ; this is known as the superconvergence property of the OSC solution. Inequalities in (29) and (31) are the maximum norm global error estimates; they imply which are optimal order error estimates. The estimate for ℎ is also optimal by regularity, whereas estimates for ℎ and ℎ are not. These results are typical for the OSC method [13].
To prove Theorem 1, we use the approach and results in [14,15]. For a given ℎ ∈ 1, , consider the differential equation subject to the interface and the boundary conditions (4c) and (4b), respectively. Using (11) and (9), get The OSC problem (18) can be written in an equivalent integral form: find ℎ ∈ 1, such that First, we prove two auxiliary lemmas. The following statement follows from Lemma 3.1 in [14].

Lemma 2.
Assume that functions , , and are in 0 , and let ∈ 0 be the unique solution of problem (10). Then the OSC problem (36) has a unique solution ℎ , and Proof. Let : 1, → 0 be defined by (12), and let be its Fréchet derivative given by (13). First note that the statement of Lemma 3.1 in [14] remains valid with space where V, is the modulus of continuity of V on the closed interval . Then an auxiliary inequality (2.1) in [14] holds over 0 .

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Under sufficient regularity assumptions on the coefficients of the differential equation, in addition to the bounds in (49), higher order derivatives of the numerical solution ℎ are locally bounded by the corresponding derivatives of the exact solution .
Proof. The statement follows immediately from Lemma 4.1 in [14].

Numerical Examples
where The exact solution of this problem is Numerical results with the maximal nodal errors the Sobolev norm errors, and the corresponding approximate convergence orders ℎ = log 2 (‖ ℎ ‖ * /‖ ℎ/2 ‖ * ) are presented in Table 1. Observe fourth-order convergence of both the solution and its derivative in the maximal nodal norm. The latter demonstrates the superconvergence property of For the case = 1 and = 0.5, numerical results are given in Table 2, and they are qualitatively similar to those in the preceding example. The machine precision is now reached at = 512.

Conclusions
The OSC method applied to IBVP (4a)-(4c) provides a highly accurate numerical solution to a problem with discontinuous coefficients. The scheme preserves the advantages of the OSC method applied to BVPs with smooth solutions. The resulting coefficient matrix has the same structure as that of the corresponding BVP without interface conditions. The obtained results can be extended to multiple interface points and to corresponding nonlinear and time dependent problems. They may also be helpful in developing OSC schemes for multidimensional interface problems.