A BIVARIATE SPLINE METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

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1 A BIVARIAE SPLINE MEHOD FOR SECOND ORDER ELLIPIC EQUAIONS IN NON-DIVERGENCE FORM MING-JUN LAI AND CHUNMEI WANG Abstract. A bivariate spline method is developed to numerically solve second order elliptic partial differential equations (PDE) in non-divergence form. he existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. Computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method. Key words. primal-dual, discontinuous Galerkin, finite element methods, spline approximation, Cordes condition AMS subject classifications. 65N30, 65N1, 35J15, 35D35 1. Introduction. We are interested in developing an efficient numerical method for solving second order elliptic equations in non-divergence form. o this end, consider the model problem: Find u = u(x) satisfying (1.1) (1.) a ij iju + cu = f, in Ω, u = 0, on Ω, where Ω is an open bounded domain in R with a Lipschitz continuous boundary Ω, ij is the second order partial derivative operator with respect to x i and x j for i, j = 1,, and the function f L (Ω). Assume that the tensor a(x) = {a ij (x)} is symmetric positive definite and uniformly bounded over Ω, the coefficient c(x) is non-positive and uniformly bounded over Ω. In addition, we assume that the coefficients a ij (x) are essentially bounded so that the second order model problem (1.1) cannot be rewritten in a divergence form. hus, the problem of designing stable and convergent numerical methods for (1.1) is subtle and currently an active area of research. See [4], [5], [6], [3], [9], and references therein. For convenience, we shall assume that the model problem (1.1) has a unique strong solution u H (Ω) satisfying the H regularity: (1.3) u H (Ω) C f L (Ω) for a positive constant C. For example, when Ω is bounded with C 1,1 smoothness boundary, the Calderon Zygmund theory (see e.g. [[14], heorem 9.15]) ensures that the solution to (1.1) has a unique solution and satisfies (1.3) if a(x) is continuous over Ω and c L (Ω). For another example, when a(x) is only L (Ω), the Cordes condition can ensure the existence and uniqueness of strong solution if the domain Ω is convex with C boundary (cf. heorem 1..1 in [0]), where the coefficient tensor a(x) is said to satisfy the Cordes condition if (1.4) a ij ( i=1 a ii) 1, in Ω Rn n 1 + ɛ Department of Mathematics, University of Georgia, Athens, GA mjlai@uga.edu. his research is partially supported by Simons collaboration grant and the National Science Foundation Award DMS Department of Mathematics, exas State University, San Marcos, exas (c w80@txstate.edu). he research of Chunmei Wang was partially supported by National Science Foundation Awards DMS and DMS

2 for a positive number ɛ (0, 1]. his Cordes condition is reasonable in R in the sense that when the coefficient tensor a(x) satisfies the standard uniform ellipticity condition, i.e., there exist two positive numbers λ 1 and λ such that (1.5) λ 1 ξ ξ ξ a(x)ξ λ ξ ξ, ξ R, x Ω, then the Cordes condition holds true in R (cf. [0]). Furthermore, the assumption that the underlying domain Ω is convex is not necessary to ensure the H regularity of the solution to the Dirichlet problem of Poisson equations. Based on the main result in [1], a bounded Lipschitz domain Ω satisfying an uniform outerball condition implies the H regularity. Here, a domain satisfies an uniform outerball condition if there exists a positive number r > 0 such that every point x on the boundary Ω, there exists a ball of radius r touched at x which lies outside of Ω. Clearly, any convex domain satisfies an uniform outerball condition with the radius r =. Also, any convex domain is a Lipschitz domain (cf. Corollary 9.1. in []). hus, the result in [1] includes convex domains as a special case. he uniform outerball condition is also called semi-convex in [1]. hus, when Ω is Lipschitz and semi-convex, there exists a strong solution u H (Ω) of (1.1) satisfying the H regularity (1.3) if the PDE coefficient tensor a(x) satisfies the Cordes condition. Next as each function a ij in the coefficient tensor a(x) is in L (Ω), we assume in this paper that a ij can be decomposed into finitely many pieces such that over each piece a ij is a continuous function. Such an assumption is reasonable as often seen in practice. Under this assumption, although using a polygonal partition may find the decomposition of a ij more conveniently, we shall use a triangulation to decompose Ω in this paper to demonstrate the numerical performance. If a polygonal mesh is indeed used, the polygonal splines constructed in [1] should be used. Recently, Smears and Süli in [4] used the well-known Lax-Milgram theorem to establish the weak solution to (1.1) with c 0. hey employed the Cordes condition to define a nonsymmetric bilinear form for their weak solution. By testing the Laplace of piecewise polynomials of degree k over a triangulation or polygonal partition, they compute their numerical solution. In fact, the solution is a strong solution due to the regularity (1.3). In [9], C. Wang and J. Wang used a primal-dual weak Galerkin finite element method to convert (1.1) with c 0 into a constrained minimization problem. he bilinear form associated with (1.1) was shown to satisfy the Lax-Babuska-Brezzi condition by using the regularity assumption (1.3). hus, the weak formulation associated with (1.1) is wellposed. he convergence and convergence rates of these two numerical methods were established in [4] and [9] together with numerical evidence of convergence over non-convex domains. In this paper, we provide another efficient computational method for numerical solution of (1.1). More precisely, we propose a bivariate spline method based on the minimization of the jumps of functions across edges and the boundary condition to solve the constrained minimization similar to the one in [9]. Bivariate splines in Sk r ( ) of smoothness r 0 and degree k > r over triangulation can be written in terms of S 1 k ( ), the space of discontinuous piecewise polynomial functions. Each polynomial over a triangle in is written in Bernstein-Bézier polynomial form (cf. [18]). he smoothness constraints across an interior edge e of triangulation are written in terms of the coefficients of polynomials over the two triangles sharing the common edge e. In particular, smoothness conditions of any order across interior edges have been implemented in MALAB which can be simply used. his is an improvement over the internal penalties in the DG method in [4] and stabilizers in the weak Galerkin method in [9]. Bivariate splines have been used for numerical solutions of various types of PDE. See [5], [19], [16], [4], [15], [], and etc.. hey can be very convenient for numerical solutions of this type of PDE. See an extensive numerical evidence in 6. Note that in [4], an hp-version discontinuous Galerkin finite element method was used. he method yielded an optimal order of convergence regarding to the mesh size h, i.e. k 1 for polynomial degree k =, 3, 4, 5. We use the C 1 spline function for the same PDE with discontinuous coefficients as in [4] and provide an evidence that the convergence rate of the root mean square error (RMSE)

3 of u S u H (Ω) using bivariate spline method is also k 1 for k =, 3, 4, 5 when c 0. In fact, our spline method produces more accurate results than that in [4] and [9]. One of the reasons is that our spline method more flexible in the sense that we can employ various spline spaces for primal and dual variables. hat is, in the primal and dual formulation, when using X h = Sk 1 ( ) for the primal variable, we can use M h = S 1 k ( ) for the dual variable instead of S 1 k ( ) and S 1 k 1 ( ) as in [9]. Such a choice can produce more accurate results although not for all the cases. (See Section 6 for detail.) In addition, we can use higher degree splines very easily by inputting a large degree in our MALAB code. he flexibility of using bivariate splines of various degrees make our method more convenient to increase the accuracy of solutions. When c 0, we have the similar convergence behavior. In particular, the convergence rate of u S u in H (Ω) semi-norm is still k 1. he paper is organized as follows: We first start with an explanation of the primal-dual discontinuous Galerkin method to solve (1.1) in the next section. Mainly, we establish some basic properties such as the existence, uniqueness, stability of the method in Section 3. hen in Section 4 we present an error analysis of the numerical solution. Next we reformulate the primal-dual discontinuous Galerkin algorithm based on the bivariate spline functions which were implemented in [5]. Extensive numerical results are reported in Section 6. We start with a PDE with smooth coefficients and test on a smooth solution to demonstrate that the bivariate spline method works very well. hen we solve some PDE with discontinuous coefficients and nonsmooth solutions. For comparison purpose, we use the PDE in (1.1) with c 0 in [4] and [9]. Although these PDEs have discontinuous coefficients and non-smooth solutions, our spline method is able to approximate the solution very well. herefore, the bivariate spline method is effective and efficient.. A Primal-Dual Discontinuous Galerkin Scheme. Our model problem seeks for a function u H (Ω) satisfying u Ω = 0 and (.1) ( a ij iju + cu, w) = (f, w), where (, ) is the standard L projection defined on the domain Ω. w L (Ω), Let h be a polygonal finite element partition of the domain Ω R. Denote by E h the set of all edges in h and E 0 h = E h \ Ω the set of all interior edges. Assume that h satisfies the shape regularity conditions described in [7, 30]. Denote by h the diameter of h and h = max h h the mesh size of the partition h. Let k 0 be an integer. Let P k ( ) be the space of polynomials of degree no more than k on the element h. For any given integer k, we define the finite element spaces composed of piecewise polynomials of degree k and k, respectively; i.e., X h = {u : u P k ( ), h }, M h = {u : u P k ( ), h }. Denote by [v ] the jump of v on an edge e E h ; i.e., 3 (.) { v 1 v [v ] =, e = ( 1 ) Eh 0, v, e Ω, where v i denotes the value of v as seen from the element i, i = 1,. he order of 1 and is non-essential in (.) as long as the difference is taken in a consistent way in all the formulas. Analogously, one may define the jump of the gradient of u on an edge e E h, denoted by [ u]. For any v X h, the quadratic functional J(v) is given by J(v) = 1 h 3 [v ], [v ] e + 1 (.3) h 1 [ v ], [ v ] e. e E h e E 0 h

4 4 It is clear that J(v) = 0 if and only if v C 1 (Ω) X h with the homogeneous Dirichlet boundary data v = 0 on Ω. We introduce a bilinear form (.4) b h (v, q) = ( a ij ijv + cv, q), v X h, q M h. he numerical solution of the model problem (1.1)-(1.) can be characterized a constrained minimization problem as follows: Find u h X h such that (.5) u h = arg min v X h, b h (v,q)=(f,q), q M h J(v). By introducing the following bilinear form (.6) s h (u, v) = e E h h 3 [u], [v ] e + e E 0 h h 1 [ v ], [ v ] e, u, v X h, the constrained minimization problem (.5) has an Euler-Lagrange formulation that gives rise to a system of linear equations by taking the Fréchet derivative. he Euler-Lagrange formulation for the constrained minimization algorithm (.5) gives the following numerical scheme. Algorithm.1. (Primal-Dual Discontinuous Galerkin FEM) A numerical approximation of the second order elliptic problem (1.1)-(1.) seeks to find (u h ; λ h ) X h M h satisfying (.7) (.8) s h (u h, v) + b h (v, λ h ) = 0, v X h, b h (u h, q) = (f, q), q M h. 3. Existence, Uniqueness and Stability. In this section, we will derive the existence, uniqueness, and stability for the solution (u h ; λ h ) of the primal-dual discontinuous Galerkin scheme (.7)-(.8). For each element h, let B be the largest disk inside of centered at c 0 with radius r and F k,b (f) be the averaged aylor polynomial of degree k for f L 1 ( ) (see page 4 of [18] for details). Note that the averaged aylor polynomial F k,b (f) satisfies (cf. Lemma 1.5 in [18]) (3.1) ijf k,b (f) = F k,b ( ijf) if ij f L1 ( ). Let P Xh (f) and P Mh (f) be interpolations/projections of f onto the spaces X h and M h defined by P Xh (f) = F k,b (f) and P Mh (f) = F k,b (f) on each element h, respectively. Using (3.1) gives rise to (3.) ijp Xh (f) = P Mh ( ijf), on each element h, Lemma 3.1. [18] he interpolant operators P Xh and P Mh are bounded in L (Ω). In other words, for any f L (Ω) we have (3.3) P Xh (f) C f, (3.4) P Mh (f) C f,

5 where denotes the L norm defined on the domain Ω, C is a constant depending only on the shape parameter θ h = max h h ρ, ρ is the radius of the largest inscribed circle of. Recall that h is a shape-regular finite element partition of the domain Ω. For any h and φ H 1 ( ), the following trace inequality holds true: (3.5) φ C(h 1 φ + h φ ). Deonte by Q k the L projection onto the finite element space M h. We introduce a semi-norm in the finite element space X h, denoted by ; i.e., ( (3.6) v = Q k ( a ij ijv + cv) + s h (v, v), v X h. he following result shows that defined in (3.6) is indeed a norm on X h when the meshsize h is sufficiently small. Lemma 3.. Assume that the H regularity (1.3) holds true for the model problem (1.1)-(1.), and that the coefficient tensor a(x) = {a ij (x)} and c(x) are uniformly piecewise continuous in Ω with respect to the finite element partition h. hen, there exists an h 0 > 0 such that in (3.6) defines a norm on X h when the meshsize h is sufficiently small such that h h 0. Proof. It suffices to verify the positivity property for. o this end, note that for any v X h satisfying v = 0 we have s h (v, v) = 0. It follows that [v ] = 0 on each edge e E h and [ u] = 0 on each interior edge e E 0 h. Hence, v C1 (Ω) and v = 0 on Ω. In addition, on each element h, we have hus, Q k ( a ij ijv + cv) = 0. a ij ijv + cv = (I Q k )( a ij ijv + cv) := F. Using the H -regularity assumption (1.3), there exists a constant C such that (3.7) v C F. Note that a ij (x) and c(x) are uniformly piecewise continuous in Ω with respect to the finite element partition h. Let ā ij and c be the average of a ij and c on each element h. hen, for any ε > 0, there exists a h 0 > 0 such that a ij ā ij L (Ω) ε, c c L (Ω) ε, if the meshsize h is sufficiently small such that h h 0. Denote by c and v the average of c and v on each element h, respectively. It follows from the linearity of the projection Q k that F a ij ā ij ijv + + (I Q k )(cv c v) Q k ((a ij ā ij ) ijv) Cε v + cv c v Cε v + (c c)v + c(v v) Cε v + Cε v + Ch v 1 Cε v + Ch v, 5

6 6 where is the H norm defined on the domain Ω, and we have used the boundedness of the L projection Q k, which, combined with (3.7), gives v C(ε + h) v. his yields that v = 0 as long as ε is sufficiently small such that Cε < 1, which can be easily achieved by adjusting the parameter h 0. his completes the proof of the lemma. We are now in a position to establish an inf-sup condition for the bilinear form b h (, ). Lemma 3.3. (inf-sup condition) Under the assumptions of Lemma 3., for any q M h, there exists a v q X h such that (3.8) (3.9) b h (v q, q) β q, v q C q, provided that the meshsize h is sufficiently small. Proof. Consider an auxiliary problem that seeks w H (Ω) H 1 0 (Ω) satisfying (3.10) a ij ijw + cw = q, in Ω. From the regularity assumption (1.3), it is easy to know that the problem (3.10) has one and only one solution, and furthermore, the solution satisfies the H regularity property; i.e., (3.11) w C q. By letting v q = P Xh (w), from (3.) we obtain ijv q = ijp Xh (w) = P Mh ( ijw). Letting ā ij be the average of a ij over h, we arrive at = = a ij ijv q + cv q {(a ij ā ij )P Mh ( ijw) + P Mh (ā ij ijw)} + (c c)p Xh (w) + P Xh ( cw) {(a ij ā ij )P Mh ( ijw) + P Mh ((ā ij a ij ) ijw) + P Mh (a ij ijw)} + (c c)p Xh (w) + P Xh (( c c)w) + P Xh (cw) = {(a ij ā ij )P Mh ( ijw) + P Mh ((ā ij a ij ) ijw)} + (c c)p Xh (w) + P Xh (( c c)w) + P Mh ( a ij ijw + cw) + P Xh (cw) P Mh (cw) =E + q + P Xh (cw) P Mh (cw). where we have used (3.10) and P Mh q = q. Here, E = {(a ij ā ij )P Mh ( ij w) + P M h ((ā ij a ij ) ij w)} + (c c)p X h (w) + P Xh (( c c)w).

7 7 With the above chosen v q as P Xh (w), we have (3.1) b h (v q, q) = ( = a ij ijp Xh (w) + cp Xh (w), q) (E, q) + q + (P Xh (cw) P Mh (cw), q). Note that the coefficient tensor a(x) = {a ij } and c(x) are uniformly piecewise continuous over h. hus, for any given suifficiently small ε > 0, we have a ij ā ij L (Ω) ε and c c L (Ω) ε for sufficiently small meshsize h. It then follows from the Cauchy-Schwarz inequality, (3.3) - (3.4), and the H regularity property (3.11) that (E, q) ( Cε ( + Cε P Xh w ( Cε ( P Mh ( ijw) ijw Cε w q Cε q, ( q q + Cε q ( + Cε ( + Cε ( w w q ( ijw ( q q and (P Xh (cw) P Mh (cw), q) ( ( P Xh (cw c w) P Mh (cw c w) q ( ( cw c w q ( (c c)w + c(w w) q (Cε w + Ch w 1 ) q C(ε + h) w q C(ε + h) q, where c and w are the average of c and w on each element h, respectively, C is a generic constant independent of h. Substituting the above estimate into (3.1) yields b h (v q, q) (1 C(ε + h)) q, which leads to the estimate (3.8) when the meshsize h is sufficiently small. It remains to derive the estimate (3.9). o this end, recall that (3.13) v q = Q k ( a ij ijv q + cv q ) + s h (v q, v q ). Letting v q = P Xh (w), the first term on the right-hand side of (3.13) can be bounded by using (3.),

8 8 (3.3) and (3.11) as follows: (3.14) Q k ( a ij ijv q + cv q ) = Q k ( C a ij ijp Xh (w) + cp Xh (w)) a ij P Mh ( ijw) + cp Xh (w) a ij L (Ω) ijw + C c L (Ω) w C q. As to the term s h (v q, v q ) in (3.13), note that it is defined by (.6) using the jump of v q on each edge e E h plus the jump of v q on each interior edge e E 0 h. For an interior edge e E 0 h shared by two elements 1 and, we have It follows that [v q ] e = v q 1 e v q e = P Xh (w) 1 e P Xh (w) e = (P Xh (w) 1 e w e ) + (w e P Xh (w) e). (3.15) [v q ], [v q ] e P Xh (w) 1 e w e e + P Xh (w) e w e e. Using the trace inequality (3.5), we have P Xh (w) 1 e w e e Ch 1 P X h (w) w 1 + Ch (P Xh (w) w) 1. Analogously, the following holds true P Xh (w) e w e e Ch 1 P X h (w) w + Ch (P Xh (w) w). Substituting the last two inequalities into (3.15) yields (3.16) [v q ], [v q ] e C i=1 For boundary edge e Ω, from w e Ω = 0 we have ( h 1 P X h (w) w i + Ch (P Xh (w) w) i ). [v q ] e = v q e = P Xh (w) e w e. hus, the estimate (3.16) remains to hold true. Summing (3.16) over all the edges yields h 3 [v q ], [v q ] e C ( h 4 P X h (w) w + Ch (P X h (w) w) ) (3.17) e E h C w where we have used the estimate (4.3) with m = 1 and s = 0, 1 in the last inequality. Combining (3.17) with the regularity estimate (3.11) gives rise to (3.18) e E h h 3 [v q ], [v q ] e C q.

9 9 A similar argument can be applied to yield the following estimate (3.19) h 1 [ v q ], [ v q ] e C q. e E 0 h We emphasize that the summation in (3.19) is taken over all the interior edges so that no boundary value for w is needed in the derivation of the estimate (3.19). Combining (3.18) and (3.19) with s h (v q, v q ) yields s h (v q, v q ) C q, which, together with (3.14), completes the derivation of the estimate (3.9). Lemma 3.4. (Boundedness) he following inequalities hold true: s h (u, v) u v, u, v X h, b h (v, q) C v q, v X h, q M h. Proof. It follows from the definition of s h (, ), and Cauchy-Schwarz inequality that for any u, v X h, we have s h (u, v) = [u], [v ] e + h 1 [ u], [ v ] e e E 0 e E h h 3 h ( ( h 3 [u], [u] e h 3 [v ], [v ] e e E h e E h ( ( + h 1 [ u], [ u] e h 1 [ v ], [ v ] e e E 0 h s h (u, u sh (v, v u v. e E 0 h Next from the definition of b h (, ),, and Cauchy-Schwarz inequality that for any v X h, q M h, we have b h (v, q) = hese complete the proof. ( = (Q k ( ( Q k ( a ij ijv + cv, q) a ij ijv + cv), q) a ij ijv + cv) ( q v q. Define the subspace of X h as follows: Ξ h = {v X h : b h (v, q) = 0, q M h }. Lemma 3.5. (Coercivity on the Kernel) here exists a constant α, such that s h (v, v) α v, v Ξ h.

10 10 Proof. For any v Ξ h, we have b h (v, q) = 0, q M h. It follows from the definition of b(, ) in (.4) that which yields 0 = b h (v, q) = ( a ij ijv + cv, q) = Q k ( (Q k ( a ij ijv + cv) = 0, on each h by letting q = Q k ( a ij ij v + cv). completes the proof with α = 1. a ij ijv + cv), q), his implies s h(v, v) = v, which Using the abstract theory for the saddle-point problem developed by Babuska [6] and Brezzi [8], we arrive at the following theorem based on Lemmas heorem 3.6. he primal-dual discontinuous Galerkin finite element method (.7)-(.8) has a unique solution (u h ; λ h ) X h M h, provided that the meshsize h < h 0 holds true for a sufficiently small but fixed parameter h 0 > 0. Moreover, there exists a constant C such that the solution (u h ; λ h ) satisfies (3.0) u h + λ h C f. 4. Error Estimates. Let (u h ; λ h ) X h M h be the approximate solution of the model problem (1.1) arsing from primal-dual discontinuous Galerkin finite element method (.7)-(.8). Note that λ = 0 is the exact solution of the trival dual problem b h (v, λ) = 0 for all v H (Ω). Define the errors functions by e h = u h P Xh u, ɛ h = λ h P Mh λ. Lemma 4.1. he error functions e h and ɛ h satisfy the following equations: (4.1) (4.) s h (e h, v) + b h (v, ɛ h ) = s h (P Xh u, v), v X h, b h (e h, p) =l u (p), p M h, where l u (p) = (a ij(i P Mh ) ij u, p) + (c(i P Xh )u, p). Proof. By subtracting s h (P Xh u, v) from both sides of (.7), we obtain s h (u h P Xh u, v) + b h (v, λ h 0) = s h (P Xh u, v), v X h, which completes the proof of (4.1).

11 11 Substracting b h (P Xh u, p) from both sides of (.8), it follows from (3.) and (1.1) that b h (u h, p) b h (P Xh u, p) =(f, p) b h (P Xh u, p) =(f, p) (a ij ij(p Xh u) + cp Xh u, p) =(f, p) =(f, p) ( (a ij P Mh ( iju) + cp Xh u, p) a ij iju + cu, p) =(f, p) (f, p) = (a ij (P Mh I) iju, p) (a ij (I P Mh ) iju, p) + which completes the proof of (4.). (a ij (P Mh I) iju, p) (c(i P Xh )u, p), (c(p Xh I)u, p) (c(p Xh I)u, p) he equations (4.1) and (4.) are called error equations for the primal-dual discontinuous Galerkin finite element scheme. his is a saddle point system for which Brezzi s heorem can be employed for the analysis of stability. Lemma 4.. [7, 30] Let h be a finite element partition of Ω satisfying the shape regular assumption given in [7, 30]. hen, for any 0 s and 1 m k, one has (4.3) h s u P Xh u s, Ch (m+1) u m+1, (4.4) h s u P Mh u s, Ch (m 1) u m 1. heorem 4.3. Assume that the coefficient tensor a(x) = {a ij (x)} and c(x) are uniformly piecewise continuous in Ω with respect to the finite element partition h. Let u and (u h ; λ h ) X h M h be the solutions of (1.1) and (.7) - (.8), respectively. Assume that the exact solution u of (1.1) is sufficiently regular such that u H k+1 (Ω). here exists a constant C such that u h P Xh u + λ h P Mh λ Ch k 1 u k+1, provided that the meshsize h < h 0 holds true for a sufficiently small, but fixed h 0 > 0. Proof. It follows from Lemmas that the Brezzi s stability conditions are satisfied for the saddle point problem (4.1)-(4.). hus, there exists a constant C such that ( ) s h (P Xh u, v) l u (p) (4.5) e h + ɛ h C + sup. v p M h,p 0 p (4.6) Recall that sup v X h,v 0 s h (P Xh u, v) sup v X h,v 0 v sup v X h,v 0 e E h h 3 [P Xh u], [v ] e + e E h 1 h 0 [ P Xh u], [ v ] e v

12 1 As to the first term of the right-hand side of (4.6), from Cauchy-Schwarz inequality, trace inequality (3.5) and (4.3), we have (4.7) ( h 3 [P ( Xh u], [v ] e C h 3 [P Xh u] e h 3 [v ] e e E h e E h e E h ( C h 3 ( [P Xh u] [u] e + [u] e) e E h ( C h 4 Ch k 1 u k+1 v, v [P Xh u u] + h [P Xh u u] 1, where we used [[u] = 0 as u H (Ω) H 1 0 (Ω). Similarly, we have (4.8) e E 0 h Substituting (4.7)-(4.8) into (4.6), we have h 1 [ P Xh u], [ v ] e Ch k 1 u k+1 v. s h (P Xh u, v) (4.9) sup Ch k 1 u k+1. v X h,v 0 v From Cauchy-Schwarz inequality and (4.4), we obtain v (4.10) sup p M h,p 0 = sup p M h,p 0 sup p M h,p 0 + sup p M h,p 0 l u (p) p (a ij(i P Mh ) ij u, p) + sup p M h,p 0 p a ij L (Ω)( (I P M h ) ij u p c L (Ω)( (I P h X h )u p (c(i P Xh )u, p) ( p ( p p Ch k 1 u k+1 + Ch k+1 u k+1 Ch k 1 u k+1. Substituting (4.9) and (4.10) into (4.5) completes the proof. 5. Bivariate Spline Implementation of Algorithm.1. We shall use a discontinuous spline space X h of degree k over a finite element partition h for the primal variable and use another discontinuous spline space M h of degree k 1, e.g. k 1 = k over h for dual variable. When h is a triangulation, these are spline spaces which have been thoroughly studied in [5] and [18]. In this paper, let us explain how to use these spline functions for numerical solution of the second order elliptic PDE (1.1). When h is a triangulation, spline functions use the Bernstein-Bézier representation as explained in [18]. hat is, the prime-dual discontinuous Galerkin FEM method discussed in the previous sections can be reformulated by using the Bernstein-Bézier representation. he representation has several nice properties (cf. [18]): (1) the basis functions form a partition of unity, () the basis functions are nonnegative, and (3) the basis functions have explicit formulas for their derivatives, integration, their inner product, and triple product integration.

13 In the remaining of the paper, we use both u X h and its coefficient vector u in terms of Bernstein-Bézier representation to write a discontinuous spline function u. Similarly, we use both q M h and its coefficient vector q. Most importantly, for any function u X h, u is a piecewise polynomial function of degree k over h, the jump function [[u] over an interior edge e of h can be rewritten by using the smoothness conditions between the coefficients of two polynomial pieces u 1 and u on their common edge e for triangles 1, h which share e. See [11] and [18]. he smoothness conditions are linear and all these conditions over each interior edge can be expressed together by using Hu = 0 as explained in [5], where H is a rectangular and sparse matrix and u is the coefficient vector of u. On the boundary of Ω, u has to satisfy the Dirichlet boundary condition which can be approximated by using a standard polynomial interpolation method, i.e., u(x) e = g(x) for k + 1 distinct points x e, where e is a boundary edge of h. As u is a polynomial on e, the interpolation condition u(x) e = g(x) can be expressed by linear equations in terms of its coefficients. We put these linear equations for all boundary edges together and express them by Bu = g, where B is a rectangular and sparse matrix and g is a vector consisting of the values of g at the k + 1 equally-spaced points over e for all boundary edges e. he PDE equation in (.1) can be discretized by using Bernstein-Bézier representation as follows. We first approximate the right-hand side f by discontinuous spline functions in S f M h. For example, we may choose S f to be the piecewise polynomial function which interpolates f at the domain points on of degree k 1 for all triangle h, under the assumption that f is a continuous function. For another example, we choose S f M h such that for each triangle h, (5.1) fqdxdy = S f qdxdy, q P k1, where P k1 is the standard polynomial space of total degree k 1. It is easy to know that the problem (5.1) has a unique solution of S f. hus, S f M h is well-defined. In fact, we have the following properties (5.) S f f and S f f = min s M h s f. Indeed, we have S f dxdy = fs f dxdy for all h and use Cauchy-Schwarz inequality to have the inequality in (5.). he equality in (5.) can be seen from the solution of the least squares problem in (5.1). We compute the inner product integration on the right-hand of (.1) exactly by using heorem.34 in [18] and a triple inner product formula. hat is, we have fqdxdy = S f qdxdy = Mf, q, Ω Ω where f is the coefficient vector of S f, M is called the mass matrix which is a blockly diagonal matrix and q is the coefficient vector of q. Similarly, we approximate the coefficients a ij by discontinuous spline functions in another discontinuous spline space S ij L h = S1 1 ( h) of degree 1, say piecewise linear interpolation of a ij. (5.3) a ij ijuqdxdy S i,j ijuqdxdy, u P k, q P k. Once we have S ij, we compute triple product integration on the left-hand side of (.1). hat is, S ij ij uqdxdy has an exact formula in terms of the coefficients of S ij, u, and q. hus we have a ij ijuqdxdy Ω S ij ijuqdxdy = Ku, q, Ω 13

14 14 where K is the stiffness matrix related to the PDE (1.1). In order to have an equality in the above formula, we now use the standard L projection P Mh which is defined by P Mh (v) M h such that (5.4) P Mh (v), q = v, q, q M h. hus, we have a ij ijuqdxdy = P ( Ω a ij iju), q = P Mh (a ij iju)qdxdy. Ω Since the the projection is linear, we can write P Mh (a ij iju)qdxdy = Ku, q, Ω for a blockly diagonal matrix K and for all q M h. In this way, we obtain a discretized PDE equation: Ku, q = Mf, q for all q R d(mh) or a linear system: (5.5) Ku = Mf. Note that both M and K can be computed in parallel. In terms of the Berstein-Bézier representation, the bilinear forms in (.6) and (.4) can be rewritten as (5.6) s(u, v) = h Hu, Hv + h Bu, Bv, u, v X h, and (5.7) b(u, q) = Ku, q, u X h, q M h. With the above preparation, Algorithm.1 can be recast as follows. Let us consider the following minimization problem for (.1): Find u satisfying (5.8) min h ( Hu + Bu g ), subject to Ku = Mf. Note that the boundary condition is imposed by minimizing the error in an least-squares sense so that the boundary conditions do not need to be strictly enforced. his minimization problem (5.8) can be reformulated by using Lagrange multiplier method as follows: let (5.9) L(u, λ) = h ( Hu + Bu g ) + λ (Ku Mf), where λ is a Lagrange multiplier. hus, the minimizer u of (5.8) satisfies (5.10). Hence, we have Algorithm 5.1. (he Primal-Dual Bivariate Spline Method) Find a vector pair (u, λ ) R d(xh) R d(mh) satisfying { h Hu, Hd + h Bu, Bd + λ, Kd = h g, Bd, d R d(xh), (5.10) q, Ku = q, Mf, q R d(mh), where d(x h ) is the dimension of X h and d(m h ) is the dimension of M h. In fact, d(x h ) = (k + 1)(k + )N( h )/ and d(m h ) = (k 1 + 1)(k 1 + )N( h )/ with N( h ) being the number of triangles in h. We shall denote by u h X h the spline solution with coefficient vector u and similarly, λ h M h with coefficient vector λ. his Algorithm 5.1 will be implemented and numerically experimented in this paper. We will have a flexibility to choose X h and M h. In [9], the researchers used k 1 = k and k 1 = k 1. We shall experiment various choices of k 1 and report our numerical results in the next section.

15 6. Numerical Results based on Minimization (5.8). We have implemented Algorithm 5.1 in MALAB based on the spline function implementation method discussed in [5] which is completely different from the spline functions implemented in []. We shall use S 1 d ( ) for d 1 over a triangulation and let S u S 1 d ( ) be the spline solution with the coefficient vector c(u) which is the minimizer of (5.8) and report the root mean squared error (RMSE) of u S u, (u S u ) = ( x (u S u), y (u S u)) and (u S u ) = 15 ( x (u S u), x y (u S u), y (u S u)) based on their values over equally-spaced points, e.g grid points located over Ω. More precisely, we report the RMSE of (u S u ) which is the average of the RMSE of x (u S u) and the RMSE of y (u S u). Similar for the RMSE of (u S u ). We shall also present the rates of convergence of RMSE between refinement levels. he remaining of this section is divided into three subsections. In the first subsection, we present numerical results based on the PDE with smooth coefficients and c 0. We also use smooth solutions to test our spline method. One of purposes is to demonstrate that our MALAB implementation is correct and is able to produce excellent numerical solution. Another purpose is to compare with the numerical results in [9]. We shall show that the higher order splines produce a much better approximation than using the lower order weak-galerkin method in [9]. In the next two subsections, we mainly present numerical results from the second order elliptic PDE with discontinuous coefficients and nonsmooth solution which were studied in [4]). Our numerical experiments show that by choosing X h = M h, the bivariate spline method, i.e. Algorithm 5.1 give a better approximation than the numerical results in [4]. Finally we show some spline solutions for PDE in (1.1) with nonzero function c for smooth and nonsmooth exact solutions. Numerical results are similar to the case when c he case with smooth coefficients. In the following examples, we shall use spline spaces S 1 d ( l) of various degrees d =, 3, 4, 5, 6, 7, 8... to solve the PDE of interest, where 0 is a standard triangulation of Ω and l is the uniform refinement of l 1 for l = 1,, 3, 4. Example 6.1. We begin with a nd order elliptic equation with constant coefficients and smooth solution u = sin(x) sin(y) which satisfies the following partial differential equation: (6.1) 3 x u + x y u + y u = f(x, y), (x, y) Ω R, where Ω is a standard square domain [0, 1] (cf. [9]). We use X h = S 1 d ( l) and M h = S 1 d ( l) with h = l. We use a triangulation 0 which consists of triangles and then uniformly refine 0 repeatedly to obtain l, l = 1,, 3, 4, 5. u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e e e e able 6.1: he RMSE of spline solutions using X h = S 1 ( l) and M h = S 1 0 ( l) for l = 1,, 3, 4, 5 of PDE (6.1) able 6.1 may be compared with able 8.1 in [9]. First of all, we recall that there is a superconvergence in L norm approximation in able 8.1 in [9]. hat is, the convergence rate in [9] is

16 16 about 4 although they only use piecewise polynomials of degree. So far there is no mathematical theory to guarantee this superconvergence. Note that the computation of their convergence is based on node points of the underlying triangulation, that is, 6 points per triangle for all triangles in h for each h > 0. In our able 6.1, the convergence is measured in the RMSE based on equally-spaced points over Ω and our convergence rate is about for M h = S0 1 ( l). Nevertheless, our convergence of (u h u) is better than that in able 8.1 in [9]. Also, we are able to show the convergence in the second order derivatives of u u h, i.e. the semi-norm u u h H (Ω). In the next few tables, we use X h = S 1 k ( l) and M h = S 1 k 1 ( l ) with k 1 1. hen the order of convergence will increase if k 1 = k. his is an advantage of our numerical algorithm over the numerical method in [9]. For k = 3 and k 1 = 1, we have u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e e e e able 6.: he RMSE of spline solutions using X h = S 1 3 ( l) and M h = S 1 1 ( l) for l = 1,, 3, 4, 5 of PDE (6.1) o increase the convergence rates for u u h and (u u h ), we use k 1 = k which can be easily adjusted in our MALAB code. As we can see from able 6.3. he convergence and convergence rates are much better than ables 6.1 and 6.. u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e e e e able 6.3: he RMSE of spline solutions using X h = S 1 3 ( l) and M h = S 1 3 ( l) for l = 1,, 3, 4, 5 of PDE (6.1) Similarly, we can use k = 4 and k 1 = 4. he numerical results are given in ables and show that the convergence rate is more than k = 4. u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e e e e able 6.4: he RMSE of spline solutions using X h = S 1 4 ( l) and M h = S 1 4 ( l) for l = 1,, 3, 4, 5 of PDE (6.1) Note that in the last row of able 6.5, the rate of convergence in L norm is 5.0 which is lower than 5.9. his is because the iterative solution of the linear system achieves the machine

17 17 u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e able 6.5: he RMSE of spline solutions using X h = S 1 5 ( l) and M h = S 1 5 ( l) for l = 1,, 3, 4, 5 of PDE (6.1) precision for this test function using MALAB. Indeed, if we use u = sin(πx) sin(πy) which is slightly harder to approximate than u = sin(x) sin(y), the rate of convergence will be around 6. See the rates of convergence in the RMSE of the spline solution shown in able 6.6, where the rate is u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e able 6.6: he RMSE of spline solutions using X h = M h = S 1 5 ( l) for l = 1,, 3, 4 of PDE (6.1) with u = sin(πx) sin(πy). u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e able 6.7: he RMSE of spline solutions using X h = M h = S 1 6 ( l) for l = 1,, 3, 4 of PDE (6.1) with u = sin(πx) sin(πy). We have tested other solutions (e.g. u = 1/(1+x +y ), u = sin(πx) sin(πy), u = sin(π(x +y )) and etc.. Numerical results are similar to ables 6.6 ables 6.8. We can see that the rate of convergence in L norm is optimal for d 5 and for sufficiently smooth solutions. hat is, the optimal convergence rate is reached when using splines in Sd 1 ( ) with d 5. Finally, our algorithm is efficient in the following sense: each table above (ables is generated within 180 seconds based on a desktop computer of 16GB in RAM with Intel Processor speed. For ables , it takes 550 seconds to generate. Major time is spent on the evaluation of spline values.

18 18 u S u rate (u S u ) rate (u S u ) rate e e e e e e e e e e e e able 6.8: he RMSE of spline solutions using X h = M h = S 1 7 ( l) for l = 1,, 3, 4 of PDE (6.1) with u = sin(πx) sin(πy). 6.. he case with discontinuous coefficients and nonsmooth solution. In this subsection, we shall demonstrate that our method works well for those PDE with discontinuous coefficients which can not be converted into its divergence form. Higher order splines can produce very accurate solutions even the solution is only C 1 (Ω). We shall use two examples studied in [4] each of which has discontinuous PDE coefficients and compare with their results to demonstrate the advantage of our bivariate spline method. Example 6.. We show the performance of our bivariate spline solutions for a PDE with discontinuous coefficients and nonsmooth exact solution u = xy(e 1 x 1)(e 1 y 1) which satisfies (6.) u + sign(x)sign(y) x x y u + u = f(x, y), y (x, y) Ω R where u = 0 on the boundary of Ω = [ 1, 1] [ 1, 1] as in [4]. As the discontinuity of one of the PDE coefficients are straight lines, we took these lines into consideration when partitioning the underlying domain as seen in Fig Note that the solution is in H (Ω), but not continuously twice differentiable. We shall use X h = S 1 d ( l) and M h = S 1 d ( l) with l shown in Figure 6.1. Instead of showing the convergence rates of u u h H (Ω) in a loglog graph for various d =, 3, 4, 5 as in [4], we present a loglog graph of the root mean squared error (RMSE) of ( D x(u u h ) + D x D y (u u h ) + D y(u u h ) )/3 based on equally-spaced points over Ω = [ 1, 1] [ 1, 1]. he graph in Fig. 6. can be compared with the one in Fig. in [4]. he comparison shows that the accuracy of our spline method is much better. One of the advantages of our method is to be able to use M h = S 1 k ( l) for various k > 0. Our experiments show that the accuracy are getting better from k = k, k 1, k, but not significantly better for k = k + 1 or larger. In order to compare with the numerical method in [9], i.e. to compare ables 8.5 and 8.6 in [9], we present a similar able 6.9 which contains the root mean square error of u u h, ( D x (u u h ) + D y (u u h ) )/, as well as ( D x(u u h ) + D x D y (u u h ) + D y(u u h ) )/3 which are based on equally-spaced points over [ 1, 1] [ 1, 1]. We can see that the accuracy of our spline solution in L norm in able 6.9 are better than those in able 8.5 and are similar to those in able 8.6 in [9]. he accuracy of our spline solution in H 1 semi-norm is much better than those in ables 8.5 and 8.6 in [9]. Higher accurate solutions are obtained when splines of higher degrees are used which can be conveniently realized by simply adjusting the degree input parameter in our MALAB code.

19 19 Fig. 6.1: riangulations l, l = 0, 1,, 3 Fig. 6.: loglog graph of RMSE ( D x(u u h ) + D x D y (u u h ) + D y(u u h ) )/3 vs the sizes of triangulations 0.5, 0.5, 0.15, 0.065, for degree d =, 3, 4, 5 (from the top to the bottom)

20 0 h = RMSE(u S u ) rate RMSE( (u S u )) rate EMSE( (u S u )) rate e e e e e e e e e e e e e e e able 6.9: he RMSE of spline solutions using the pair X h = S 1 ( l), M h = S 1 ( l) of spline spaces for l = 0, 1,, 3, 4 of PDE (6.) based on uniform triangulations in Figure 6.1 RMSE(u S u ) rate RMSE( (u S u )) rate EMSE( (u S u )) rate e e e e e e e e e e e e e e e able 6.10: he RMSE of spline solutions using the pair X h = S 1 3 ( l), M h = S 1 3 ( l) of spline spaces for l = 0, 1,, 3, 4 of PDE (6.) based on uniform triangulations in Figure 6.1 RMSE(u S u ) rate RMSE( (u S u )) rate EMSE( (u S u )) rate e e e e e e e e e e e e e e e able 6.11: he RMSE of spline solutions using the pair X h = S 1 4 ( l), M h = S 1 4 ( l) of spline spaces for l = 0, 1,, 3, 4 of PDE (6.) based on uniform triangulations in Figure 6.1 In able 6.1, we note that the RMSE(u S u ) gets deteriorated in the last refinement which indicates that the machine accuracy is achieved and the result could not be improved although the RMSE( (u S u )) still improves at the expected convergence rate. Furthermore, when using the degree of splines d 6, such a deterioration of iterations continues as the accuracy of the spline coefficient vectors could not be achieved less than 1e 15 and thus, the RMSE(u S u ) could not be better than 1e 1. In order to show the rate of convergence when d 6, the computation needs a triple or quadruple precision which will be left for a future study.

21 1 RMSE(u S u ) rate RMSE( (u S u )) rate EMSE( (u S u )) rate e e e e e e e e e e e e e e e able 6.1: he RMSE of spline solutions using the pair X h = S 1 5 ( l), M h = S 1 5 ( l) of spline spaces for l = 0, 1,, 3, 4 of PDE (6.) based on uniform triangulations in Figure 6.1 Example 6.3. In this example, we study the numerical solution to the following (6.3) (1 + x x + y ) x u + xy x + y x y u + (1 + y x + y ) y u = f(x, y), (x, y) Ω R xy x +y where Ω = (0, 1), u = (x + y ) α/. Note that the middle coefficient fails to be continuous at one corner of Ω. his PDE has been studied in [14] and [0] to explain the possibility of ill-posedness of the problem. In [4] and [9], two numerical methods find a good approximation of the solution. We shall apply our spline method to find approximations of the exact solution when α = 1.6 as in the previous literature. First, we use standard uniform refinements of a simple triangulation of Ω by adding two diagonals. RMSE(u S u ) rate RMSE( (u S u )) rate EMSE( (u S u )) rate e e e e e e e e e e e e e e e able 6.13: he RMSE of spline solutions using the pair X h = S 1 ( l), M h = S 1 0 ( l) of spline spaces for l = 0, 1,, 3, 4 of PDE (6.3) based on uniform refinements of a simple triangulation We can see that although the accuracy of our spline solution in L norm in able 6.13 are not as good as those in ables 8.7 and 8.8 in [9], the accuracy of our spline solutions in H 1 semi-norm is better. In [4], Smears and Süli provided a numerical method to be able to achieve the convergence in an exponential decay fashion by designing a sequence of quadrilateral partitions. In this paper, we provide a simple approach to improve the numerical solution of the PDE (6.3) by starting with a special triangulation in Fig. 6.3 since the solution at one of the corners of Ω is singular and then uniformly refine it to obtain a sequence of triangulations. Over such a sequence of triangulation, numerical results from our spline method in able 6.14 are much better than those in [9], and better than the one [4] in H semi-norm although we use much more elements and degrees of freedom. We have tested our spline method for numerical solution of (6.3) for various degrees of splines for primal and/or dual variables. We do not report the results here due to the space limitation.

22 RMSE(u S u ) rate RMSE( (u S u )) rate EMSE( (u S u )) rate e e e e e e e e e e e e e e e e e e able 6.14: he RMSE of spline solutions using the pair X h = S 1 ( l) and M h = S 1 0 ( l) of spline spaces for l = 1,, 3, 4, 5, 6 of PDE (6.3) based on uniform refinements of a fixed triangulation Fig. 6.3: A fixed triangulation 6.3. Numerical Results of PDE in (1.1) with nonzero c. In this subsection, we present some numerical results from our bivariate spline method for numerical solution of the PDE in (1.1) with nonzero c. We use three examples to demonstrate that our method is effective and efficient no matter the PDE coefficients are smooth or not smooth and the solutions are smooth or not so smooth. Example 6.4. We begin with a nd order elliptic equation with smooth coefficients and smooth solution u = sin(πx) sin(πy) which satisfies the following partial differential equation: (6.4) 3 x u + x y u + y u (1 + x + y )u = f(x, y), (x, y) Ω R, where Ω is a standard square domain [ 1, 1] which is split into 4 equal sub-squares and each subsquare is split into triangles to form an initial triangulation 0. Let l be the lth uniform refinement of 0. Example 6.5. In this example, we use our spline method to solve the following PDE with discontinuous coefficients, but smooth solution. (6.5) a(x, y) u + b(x, y) u + c(x, y) x x y y u (1 + x + y )u = f(x, y), (x, y) Ω R where a(x, y) = 1 + x, b(x, y) = (xy/3, c(x, y) = 1 + y and Ω is a standard domain [ 1, 1]. We use u = sin(πx) sin(πy) as the exact solution. he same triangulations l as in Example 6.4 will

23 3 RMSE(u S u ) rate RMSE( (u S u )) rate RMSE( (u S u )) rate e e e e e e e e e e e e able 6.15: he RMSE of spline solutions using the pair X h = S 1 5 ( l), M h = S 1 3 ( l) of spline spaces for l = 1,, 3, 4 of PDE (6.4) be used. RMSE(u S u ) rate RMSE( (u S u )) rate RMSE( (u S u )) rate e e e e e e e e e e e e able 6.16: he RMSE of spline solutions using the pair X h = S 1 5 ( l), M h = S 1 5 ( l) of spline spaces for l = 1,, 3, 4 of PDE (6.5) Example 6.6. In this example, we show the performance of our spline solutions for a PDE with discontinuous coefficients and nonsmooth exact solution u = xy(e 1 x 1)(e 1 y 1) which satisfies (6.6) u + sign(x)sign(y) x x y u + y u (1 + x + y )u = f(x, y), (x, y) Ω R where u = 0 on the boundary of Ω = [ 1, 1] [ 1, 1] as in [4]. Note that the solution is in H (Ω), but not continuously twice differentiable. he same triangulations l as in Example 6.4 were used and S 1 5( l ) were used to solve the PDE in (6.6). he RMSE for spline approximation to the exact solution is shown in able RMSE(u S u ) rate RMSE( (u S u )) rate RMSE( (u S u )) rate e e e e e e e e e e e e able 6.17: he RMSE of spline solutions using the pair X h = S 1 5 ( l), M h = S 1 5 ( l) of spline spaces for l = 1,, 3, 4 of PDE (6.6) REFERENCES [1] V. Adolfsson, L integrability of second order derivatives for Poisson equations in nonsmooth domain, Math. Scan. 70(199), [] M. S. Agranovich, Sobolev Spaces, heir Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics. Springer, Cham, 015. [3] G. Awanou, Robustness of a spline element method with constraints. J. Sci. Comput., 36(3):41 43, 008. [4] G. Awanou, Spline element method for Monge-Ampre equations. BI 55 (015), no. 3, [5] G. Awanou, M. -J. Lai, and P. Wenston. he multivariate spline method for scattered data fitting and numerical solution of partial differential equations. In Wavelets and splines: Athens 005, pages Nashboro Press, Brentwood, N, 006.