ANALYSIS OF A MODIFIED SCHRÖDINGER OPERATOR IN 2D: REGULARITY, INDEX, AND FEM

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1 ANALYSIS OF A MODIFIED SCHRÖDINGER OPERATOR IN 2D: REGULARITY, INDEX, AND FEM HENGGUANG LI AND VICTOR NISTOR Version: 2.; Revised: April 2 by V.; Run: January 4, 2008 Abstract. Let r = (x 2 +x2 2 )/2 be the distance function to the origin O R 2, and let us fix δ > 0. We consider the Schrödinger type mixed boundary value problem u + δr 2 u = f H m (Ω) on a bounded, curvilinear polygonal domain Ω R 2. The singularity in the potential δr 2 severely limits the regularity of the solution u. This affects the rate of convergence to u of the finite element approximations u S S obtained using a quasiuniform sequence of meshes. If the edges of Ω are straight lines, we show that a suitable graded sequence of meshes recovers the quasi-optimal convergence rate u u n H (Ω) C dim(sn) m/2 f H m (Ω), where Sn are the FE spaces of continuous, piecewise polynomial functions of degree m associated to our sequence of meshes and u n = u Sn S n are the FE approximate solutions. This is in spite of the fact that u H (Ω) in general. One of the main results of our paper is to show that the singularities due to the potential and the singularities due to the singularities of the domain or to the change in boundary conditions can be treated in the same way. Our proof is based on regularity and well-posedness results in weighted Sobolev spaces, with the weight taking into account all singularities (including the ones coming from the potential). Our regularity results apply also to operators with weaker singularities, like the Schrödinger operator u + δr, for which we also obtain Fredholm conditions and a formula for the index. Contents. Introduction 2. Weighted Sobolev Spaces 4 3. Well-posedness and Regularity 9 4. Estimates for the Finite Element Method 5 5. Numerical Results 22 References 26. Introduction Let Ω R 2 be a bounded curvilinear polygonal domain. We assume that a finite set of points S Ω was fixed and let R(x) := Q S x Q 2, where x Q denotes the Euclidean distance from x to Q. We also assume that a smooth function V.N and H.Li were supported by NSF Grant DMS , DMS , and OCI

2 2 H. LI AND V. NISTOR V : Ω S R is given such that the product RV extends to a smooth function on Ω. We define () H := + V. The typical example we have in mind is H = + δr 2, where δ > 0 and r(x) := x O = x is the distance from x to the origin O. In this typical example, S consists of a single point, S = {O}, O Ω. The origin O is not required to be a vertex of Ω. We are interested in studying the Finite Element approximations of the solutions to the Schrödinger type mixed boundary value problem (2) Hu = f in Ω, ν u = 0 on N Ω, and u = 0 on D Ω, for a disjoint decomposition of the boundary Ω = D Ω N Ω into a region with Dirichlet and, respectively, Neumann boundary conditions. Equations of this kind appear in Quantum mechanics in the form of Schrödinger equations with centrifugal potentials [50, 49] and in fluid dynamics [43]. Therefore, the study of the regularity and of the numerical approximation of the solutions of our Schrödinger type mixed boundary value problem, Equation (2), are of practical importance. Non-homogeneous boundary conditions can also be treated by reducing to homogeneous boundary conditions. The case when V is non-singular is well understood, so we shall mainly concentrate on the case when V has non-trivial singularities of the form δr 2. In this case, the usual theorems on the well-posedness of elliptic boundary value problems in the usual Sobolev spaces [29, 42, 52, 60] do not apply. In fact, the solution u of Equation (2) will have limited Sobolev regularity near the points Q S, near the vertices of Ω, or near the points where the type of the boundary conditions changes. This is an issue, because, as is well known, the lack of regularity of the solution u slows down the convergence rate for the numerical approximation in the Finite Element Method when quasi-uniform meshes are used. It turns out, however, that the difficulties caused by the singularities of our potential V are of the same nature as the singularities caused by the vertices of Ω or by the presence of mixed boundary conditions. This observation is the starting point for the work presented in this paper. Many papers were devoted to the analysis of the singularities of (2) arising from the non-smoothness of the boundary when the potential V is smooth, see for example the monographs [26, 3, 35, 36, 5] and the references within. See also the research papers [5, 3, 7, 24, 25, 33, 34, 44, 45, 46, 54, 62]. The numerical approximation of the solutions of (2) for V = 0 was studied in a very large number of papers, including [3, 4, 8,, 2, 24, 27, 53, 56]. The case of the Schrödinger operator with a magnetic potential on a polygonal domain was studied in [6]. A good introduction to the various methods for treating corner singularities in the plane can be found in [59]. Significantly less papers were devoted to the case of singular coefficients. Nevertheless, techniques for the estimation of the finite element approximation for boundary value problems with singular coefficients can be found in the papers of Eriksson and Thomée [28], Franchi and Tesi [30], Li [39], and in the references

3 MODIFIED SCHRÖDINGER OPERATOR AND FEM 3 therein. Also, Bespalov and Rukavishnikov [5, 57] studied the p-version finite element approximation in the case when V has a single singularity at the origin, if the origin is a boundary point. Arroyo, Bespalov and Heuer [7] investigated the finite element method in some low-order weighted space for equations with singular coefficients of a different type than ours. These approaches to approximating solutions of partial differential equations with singular coefficients are thus seen to depend on the characters of the singularities. In this paper, we provide a unified numerical treatment of the difficulties caused by the singularities of the coefficients, the geometry of the domain, and the boundary conditions in the framework of weighted Sobolev spaces. These weighted Sobolev spaces are defined to take into account all the singularities (introduced by the domain, by the change of boundary conditions, and by the singularities of V ). In order to introduce these weighted Sobolev spaces, let us first notice that the assumption that RV extends to a smooth function on Ω continues to be satisfied if we increase the set S of singular points (used to define R := Q S x Q 2 ). We shall assume therefore from now on that S also contains all vertices of Ω and all points where the boundary conditions change from Dirichlet to Neumann. Let ϑ(x) be the distance from x to S. Also, let m N {0}, N := {, 2,...}, and a R. Then we define the mth weighted Sobolev space on Ω with index a by (3) K m a (Ω) := {u : Ω R, ϑ i+j a i x j yu L 2 (Ω), i + j m}. One can allow the index a to be different at every point of S, as in [26, 35, 40, 54] for instance, but we shall not pursue this simple generalization in order not to complicate the notation. Let us denote by (v, v 2 ) the L 2 -inner product of two functions v, v 2. Also, let us introduce the space V of our weak formulation by (4) V := K (Ω) {u = 0 on DΩ}. The weak solution u V of our Schrödinger type mixed boundary value problem, Equation (2), is then defined by (5) a(u, v) := ( u, v) + (V u, v) = (f, v), v V. (See Equation (7) for the case of non-homogeneous boundary conditions.) We first establish in Theorem 3. regularity results in the weighted Sobolev spaces K m a (Ω) for the weak solution u of Equation (2). In particular, we obtain that there is no loss of K m a (Ω)-regularity for the solution u. Under some additional assumptions on the potential V and on the boundary conditions, we also establish the wellposedness of the boundary value problem (2), that is, we establish the existence of a unique solution u depending continuously on the data f. See Babuska and Aziz [9], Bacuta, Nistor, and Zikatanov [3], Costabel and Dauge [24], Kondratiev [34], Lubuma and Nicaise [44], Mazzucato and Nistor [46], Nicaise [54] or the monographs [26, 35, 36, 5] for related results. By analogy with the definition of the weak solution of (2), the discrete solution u S S V := K (Ω) {u = 0 on DΩ} of (2) is defined as usual by (6) a(u S, v S ) := ( u S, v S ) + (V u S, v S ) = (f, v S ), v S S. Assume now that Ω is a polygonal domain P with straight edges. Using our wellposedness results on Equation (2), we shall exhibit a simple, explicit construction of a sequence of meshes T n on P, suitably graded towards each singular point Q

4 4 H. LI AND V. NISTOR S that provides quasi-optimal rates of convergence for the finite element method applied to H := + V in the following sense. Let S n V be the sequence of Finite Element Spaces consisting of continuous functions on P that coincide on each triangle of the mesh T n with a degree m polynomial and vanish on D P S. Let u n := u Sn S n be the corresponding discrete solutions. Then (7) u u n K (P) C dim(s n ) m/2 f H m (P), where f H m (P), m, is otherwise arbitrary and C is a constant that depends on P and m, but not on n or f (we do not assume u H (P)). Therefore we recover the optimal rate of convergence that is expected for smooth solutions [6, 9, 2, 59]. This is the content of Theorem 4., which is the main result of the paper. The rest of this paper is organized as follows. In Section 2, we introduce the weighted Sobolev spaces Ka m (Ω) used in this paper and study their properties. We also introduce in this section notation that will be used throughout this paper. In Section 3, we prove our theoretical results on the regularity and well-posedness of the solution u of Equation (2) in the spaces Ka m (Ω). Moreover, we shall prove that, under certain additional mild assumptions on the potential V, the operator H = + V is Fredholm (8) H : K a+ (Ω) {u DΩ = 0} { ν u NΩ = 0} K m a (Ω) with the exception of a in a certain countable subset. Moreover, for a small enough (and under a few mild additional assumptions), we show that H is an isomorphism. This isomorphism is crucial for the construction of finite subspaces to obtain the quasi-optimal rates of convergence. Beginning with Section 4, we assume that our domain is a straight polygonal domain P. We then construct our sequence of meshes T n, suitably graded towards the vertices, and we analyze the finite element solutions u n = u Sn defined by the variational form, Equation (6). The main result of this paper is to prove that the sequence u n satisfies quasi-optimal rates of convergence, namely Equation (7), see Theorem 4.. In Section 5, we present numerical tests on some model problems on different straight polygonal domains. We assume for simplicity that S consists of a single singular point Q of V, and we consider both the cases when Q is an internal point or a point on the boundary. For both cases, we compare the rates of convergence of the numerical solutions on different meshes. Our tests suggest quasi-optimal rates of convergence and are in complete agreement with our theoretical results. Acknowledgments. We would like to thank Anna Mazzucatto and Jorge Sofo for useful discussions. Also, we are grateful to James Brannick and Ludmil Zikatanov for suggestions on the numerical implementation of the model problem. 2. Weighted Sobolev Spaces Throughout this paper, by Ω R 2 we shall denote a bounded polygonal domain with possibly curved edges. For simplicity, we do not consider cracks or other pathologies, although they could be treated as in this paper, using also the techniques of [26, 40]. As we have explained in the Introduction, it is more convenient to consider the Schrödinger type mixed boundary value problem (2) in weighted Sobolev spaces.

5 MODIFIED SCHRÖDINGER OPERATOR AND FEM 5 In this section, we shall introduce the weighted Sobolev space Ka m (Ω) and establish their needed properties. More results on Sobolev spaces with weights can be found in [,, 26, 34, 35, 36, 37, 55]. In this paper, we shall consider only real Hilbert spaces. 2.. Notation. We adopt the standard notation for the usual, real valued Sobolev space H m (Ω) on Ω, [29], for m N {0}, N := {, 2,...}, H m (Ω) = {v : Ω R, α v L 2 (Ω), α m}, with the multi-index α = (α, α 2 ) Z 2 +, α = α + α 2 and α = α α2. We denote v 2 L 2 (Ω) = Ω v2 dxdy, as usual. The H m -norm for v H m (Ω) is defined by v 2 H m (Ω) := α v 2 L 2 (Ω). α m Let us now review the definition of the weighted Sobolev spaces used in this paper. Recall that the definition of our weighted Sobolev spaces depends on the choice of a subset S = {Q 0,..., Q v } Ω, as in the Introduction. Also, recall that we assume that the set of vertices of Ω and the set of points where the boundary conditions change are both contained in S. In the following definition of the weighted Sobolev spaces K m a (Ω), we want to replace the weight ϑ with a smoothed version, denoted by ρ. The following remark explains how this is done. Remark 2.. Let us denote by l the minimum of the non-zero distances from a point Q S to an edge of Ω. Let (9) l := min(/2, l/4) and Vi := Ω B(Q i, l), where B(Q i, l) denotes the ball centered at Q i S with radius l. Note that sets V i := Ω B(Q i, l) are disjoint. Also, we denote by O i = V i /2 := Ω B(Q i, l/2). Then, we define (0) ρ : Ω [0, 2 l], ρ C ( Ω S) { ϑ(x) = x Q i on V i and ρ(x) = ρ(x) l/2 outside V := V i. The quotients ρ/ϑ and ϑ/ρ are bounded, and we can replace ϑ with ρ in all the definition of the weighted Sobolev spaces. In particular, this leads to the following equivalent definition of the weighted Sobolev spaces: Definition 2.2. We define () K m a (Ω) := {v : Ω R, ρ i+j a i x j yv L 2 (Ω), i + j m}. For any open set G Ω and any v : G R, we denote v 2 K m a (G) := ρ α a α v 2 L 2 (G). α m We endow Ka m(ω) with the norm v 2 K m a (Ω). In particular, the inner product on the Hilbert space Ka m (Ω) becomes (u, v) K m a (Ω) = ρ 2(i+j a) ( x i yu)( j x i yv)dxdy. j i+j m Ω

6 6 H. LI AND V. NISTOR The completeness of L 2 (Ω) and standard arguments, see [29] for instance, show that the weighted Sobolev space K m a (Ω) is complete. All the resulting norms are equivalent to the ones obtained from Equation (3), in which we first introduced the weighted Sobolev spaces using the weight ϑ. (Our definitions and results extend right away to complex Hilbert spaces, but that would require us to use sesqui-linear forms instead of bilinear forms and would complicate the presentation.) 2.2. Some lemmas. In this subsection, we shall summarize several properties of Ka m that are useful for the development of our theory in Section 3 and Section 4. As usual, we shall use r and θ as the variables in the polar coordinates. In addition, by A B, we mean that there exist constants C > 0, C 2 > 0, such that C A B C 2 A. Below, by an isomorphism of Banach spaces we shall mean a continuous bijection. We first have the following alternative definition of the space Ka m(ω). Let l := min(/2, l/4), V i := Ω B(Q i, l), and O i = V i /2 := Ω B(Q i, l/2) be as in Equation (9) of our previous remark. For simplicity, we shall denote from now on Ka m := Km a (Ω) and Hm := H m (Ω). Lemma 2.3. On every V i := Ω B(Q i, l), we set a local polar coordinate system (r, θ), where Q i is sitting at the origin (and hence ρ = r on each V i ). Denote O i = V i /2 = Ω B(Q i, l/2), as before, and O := O i. Then, the weighted Sobolev space K m a := K m a (Ω) is also given by K m a = {v : Ω R, v Hm (Ω O), and r a (r r ) j k θ v L2 (V i ), j + k m}. Proof. We note that Ω O is a subset of Ω, whose closure is away from all Q i S. According to the definitions of ρ and l, ρ l/2 on Ω O. Thus, the H m -norm and the Ka m -norm are equivalent on this region, since ρ is smooth and bounded from above and from 0. On the region V i, the differential operators x and y can be written in terms of r and θ, the variables in the polar coordinates centered at Q i, x = (cosθ) r sin θ r θ Since ρ = r on V i, we have ρ j+k a x j k y u L 2 (V i) j+k m = j+k m y = (sin θ) r + cosθ θ. r r j+k a (cosθ r sin θ r θ) j (sin θ r + cosθ θ ) k u r L 2 (V i) r a (r r ) j θ k u L 2 (V i). j+k m We have a similar formula expressing r r and θ in terms of r x and r y, which provides us with the opposite inequality and completes the proof. Lemma 2.4. The function ρ j+k a j x k yρ a is bounded on Ω.

7 MODIFIED SCHRÖDINGER OPERATOR AND FEM 7 Proof. The function ρ j+k a x j k y ρa is bounded on Ω O := Ω ( O i ) because ρ is smooth and l/2. It remains to verify that our function is bounded on every V i O i as well. By changing x, y into r, θ in the polar coordinates centered at Q i S, as in Lemma 2.3, we have ρ j+k a x j k y ρa = r j+k a (cosθ r sinθ r θ) j (sin θ r + cosθ θ ) k r a r C r a (r r ) s θr t a s+t j+k C r a s j+k (r r ) s r a = C Therefore, ρ j+k a j x k y ρa is bounded on Ω, as stated. s j+k This lemma leads to the following isomorphism between weighted Sobolev spaces. Lemma 2.5. We have ρ b K m a = K m a+b, where ρb K m a = {ρ b v, v K m a }. More precisely, the multiplication by ρ b defines an isomorphism K m a Km a+b. Proof. Let v Ka m and w = ρb v. Then ρ i+j a x i j y v L2, for i + j m. Thus, we verify w Ka+b m by checking the inequalities below, ρ i+j a b x i yw j ( )( ) i j = ρ i+j a b s s t x yρ t b x i s y j t v s i,t j C ρ (i+j s t) a x i s j t v L 2 (Ω), s i,t j with the last inequality following from Lemma 2.4. Therefore, ρ b Ka m is continuously embedded in Ka+b m. In other words, the map ρb : Ka m Ka+b m is continuous. On the other hand, because this embedding holds for any real number b, we have the opposite inclusion: Ka+b m = ρ b ρ b Ka+b m ρ b Ka m. To complete the proof we also notice that the inverse of multiplication by ρ b is multiplication by ρ b, which is also continuous. Recall that V i := Ω B(Q i, l) is a small neighborhood of Q i S. Therefore, ρ(x, y) l on V i, and we have Lemma 2.6. Let G V i be an open subset of V i, such that ρ ξ l on G. Then, for m m and a a, we have (a) Ka m Ka m := Km a (Ω) and (b) u K m a (G) ξa a u K m a (G), u Ka m. Proof. Direct calculation. Let G Ω S be a compact subset of Ω. Note that there exists a constant C > 0, such that ρ C on G. Then, the following lemma asserts that the H m -norm and the K m a -norm are equivalent on H m ( G). Recall that l and V i were introduced in Equation (9). y a s.

8 8 H. LI AND V. NISTOR Lemma 2.7. For 0 < ξ l/4, let G Ω be an open subset, such that ρ ξ on G. Then, u H m ( G) M u K m a ( G) and u K m a ( G) M 2 u H m ( G), for u Hm ( G). M and M 2 depend on ξ and m, but not on u. Proof. We note that ρ is smooth and bounded below by ξ on G. Then, the proof follows the definitions of Ka m and Hm. We have the following simple comparison property of K m a (V i ) and H m (V i ). Lemma 2.8. Let G V i be an open subset, such that ρ ξ l on this region. Let m 0, then (a) u H m (G) ξ a m u K m a (G), if a m, and (b) u K m a (G) ξ a u H m (G), if a 0. Proof. A direct calculation using Lemma 2.6. We shall need the extension of Lemma 2.8 to the entire domain Ω, which reads as follows. Recall that we denote H m := H m (Ω) and K m a := Km a (Ω). Corollary 2.9. Let u : Ω R, then u H m M u K m m and u K m a M 2 u H m for a 0, where M and M 2 depend on m and a. Proof. The proof is a direct combination of the estimates in Lemma 2.7 and the estimates for the neighborhoods of all Q i S in Lemma 2.8. Recall the function R(x) = Q S x Q 2 and the operator H := + V from the introduction. Also, recall that the function V is such that the product R(x)V (x) extends to a smooth function on Ω. Using this, we can conclude this section with the following result. Lemma 2.0. The map H : Ka+ := Ka+ (Ω) Km a continuous for all m. Proof. We shall show that there is C > 0 such that Hu K m a is well defined and C u K a+ for all u K a+. Indeed, on Ω O := Ω ( i O i ), H = + V is a second order differential operator with bounded coefficients, because V C < on Ω O. Therefore, it defines a bounded operator H (Ω O) H m (Ω O) [29]. Lemma 2.7 then gives Hu K m a (Ω O) C u K m a+ (Ω O). On V i, we write + V = r 2( (r r ) θ ) + r2 V ). Then Lemma 2.3 and the equation r λ (r r ) n r λ u = (r r + λ) n u give Hu K m a (V i) C u K m a+ (V i). Adding all the similar inequalities completes the proof. Let us say a few words about the boundary weighted Sobolev spaces (2) K m a (S) F SK m a (S), where S Ω is a union of closed sides of Ω. Each side can be identified with I = [, ], so it is enough to define (3) K m a (I) = {( t2 ) k a f (k) L 2 (I), k m}. For s [0, ) we define K s a (I) by interpolation, and for s < 0 we extend this definition by duality: K s a(i) = (K s a(i)). See [, 2, 3, 46, 37, 6].

9 MODIFIED SCHRÖDINGER OPERATOR AND FEM 9 It was shown that (4) Ka m := Km a (Ω) u u Ω K m /2 a /2 ( Ω), m, is a continuous surjective map, a result that generalizes the usual result corresponding result for smooth, bounded domains. A similar result holds for the normal derivative, yielding again a continuous and surjective map (5) Ka m (Ω) := Km a (Ω) u νu Ω K m 3/2 a 3/2 ( Ω), m 2. See for example [, 2, 3, 37, 46, 6]. Note that, although it is enough to consider only integral Sobolev spaces in the interior of Ω, we need also fractional Sobolev spaces in the interior. Fortunately, the usual issues with Sobolev spaces of fractional order of the form integer+/2 do not arise in the case of the Sobolev spaces on the boundary Ω (they do arise though for the ones on Ω). See [42] for a discussion of these issues in the case of smooth boundaries. 3. Well-posedness and Regularity In this section, we shall study the well-posedness and the regularity of solutions of the Schrödinger type mixed boundary value problem (2) in the weighted Sobolev spaces Ka m := Km a (Ω). 3.. Assumptions. The following general assumptions will be satisfied throughout this paper (some of these assumptions have already been mentioned). The reader will be reminded of these assumptions when appropriate. General Assumptions. The set S Ω is a fixed, finite set containing the vertices of Ω. We assume that the boundary of Ω was partitioned as a disjoint union Ω = D Ω N Ω, with D Ω a closed set consisting of the union of finitely many smooth curves without self-intersection and with end points contained in S. We also fix the polynomial R(x) := Q S x Q 2 and a smooth function V : Ω S R such that RV extends to a smooth function on Ω. The set D Ω in our General Assumption corresponds to the part of the boundary where we impose Dirichlet boundary conditions, whereas the set N Ω corresponds to the part of the boundary where we impose Neumann boundary conditions. This allows us to consider the following slight extension of our Schrödinger type mixed boundary value problem, Equation (2), namely, to consider the mixed Boundary Value Problem Hu = f in Ω, (6) ν u = g N on N Ω, and u = g D on D Ω, where, we recall, H = + V. Our General Assumption thus shows that the points where the boundary conditions change are contained in S. We shall make the following simplifications of the mixed Boundary Value Problem (6). First, by the surjectivity of the trace map Ka m Km /2 a /2 ( Ω), we can assume g D = 0. Then, with integration by parts, we actually look for a minimal regularity solution u V := K(Ω) {u = 0 on D Ω} of (6) defined in the following weak sense: (7) a(u, v) := ( u, v) + (V u, v) = (f, v) L 2 (Ω) + g N, v NΩ,

10 0 H. LI AND V. NISTOR for all v V, where g N, v DΩ is the value of the distribution g N on the function v. (Recall that we only consider real Hilbert spaces.) Equation (7) extends the Equation (5) in the Introduction by introducing the Neumann boundary condition ν u = g N on N Ω in the definition of u Regularity. We have the following standard regularity result. Theorem 3.. Let our General Assumptions be satisfied. Let u Ka+ := Ka+(Ω) be a solution of Equation (6) with Hu := u + V u = f Ka m and with boundary data g D K /2 a+/2 ( DΩ) and g N K m /2 a /2 ( NΩ), m. As always, we suppose that our general assumptions above are satisfied. Then u Ka+ and we have u K a+ C reg ( f K m a + g D /2 K a+/2 ( DΩ) + g N m /2 K a /2 ( NΩ) + u ) K 0 a+, for a constant C reg = C reg (m, a) > 0 independent of f, g D, g N, and u. (The relation ν u = g N is defined in a weak sense after we subtract a suitable function that will make u = 0 on D Ω, as in Equation (7).) Proof. This result is standard, so we include only a sketch. Regularity is a local property, so it is enough to restrict our study close to the points of S where V may have singularities. Close to a vertex Q of Ω where V is non singular, the result is well known, see [4, 3, 24, 34, 35, 46, 5]. Assume, for simplicity that Q = O, the origin. A simple proof is obtained then by using a radial partition of unity of the form φ n (x) := φ 0 (2 n x) and then applying to the functions φ n u the usual regularity results for smooth domains. Details of this method can be found in [, 24, 35, 46], where further references are provided. It remains to deal with the behavior of u near a point Q S where V is singular. If Q is on the boundary or a vertex, then the proof is exactly the same as for the case when Q is vertex and V is non-singular at Q. When Q is an interior point of Ω, the proof is again very similar to the case when Q is a vertex, but this time one uses the simpler elliptic regularity results on R 2 applied to the functions φ n u. (One can also think of this case as using periodic boundary conditions.) This regularity result for H gives right away by induction the following regularity result for eigenfunctions, which we hope will be useful for the numerical determination of the eigenvalues and eigenfunctions of H. Corollary 3.2. Assume u Ka 2 := Ka(Ω), 2 a 0, is an eigenfunction of H (i.e., Hu = λu) such that u = 0 on D Ω and ν u = 0 on N Ω. Then u Ka m for any m N Well-posedness. Now, we shall prove the existence and uniqueness of the solution of the mixed Boundary Value Problem (2) in weighted Sobolev spaces. (This is the well-posedness of this problem in the sense of Hadamard.) The wellposedness is not satisfied, in general, if only the General Assumptions are satisfied, so we need to include a few hypothesis on V in the following theorem. Theorem 3.3. Assume, in addition to our General Assumptions, that V 0 on Ω and that V R(Q) > 0 if Q S Ω. Also, we assume that if Q S separates two adjacent sides of Ω that are assigned Neumann boundary conditions, then again V R(Q) > 0. Assume also that D Ω is not empty or that V > 0 at some point of

11 MODIFIED SCHRÖDINGER OPERATOR AND FEM Ω. Then there exists η > 0 such that, for any f Ka m, g D K /2 a+/2 ( DΩ), and g N K m /2 a /2 ( NΩ), m 0, a < η, the mixed Boundary Value Problem (6) has a unique solution u K a+ u K a+ := K a+ C m,a ( f K m a (Ω), which satisfies + g D K /2 a+/2 ( DΩ) + g N K m /2 a /2 ( NΩ) ), for a constant C m,a > 0 independent of f, g D, and g N. (For m = 0, we assume g D = 0 and we interpret the equation ν u = g N in a weak sense.) Proof. We can assume g D = 0. We shall prove first the case m = 0. This case will follow from the Lax Milgram Lemma and the strict positivity (or strict coercivity) of H on the space V := K (Ω) {u = 0 on DΩ} (or more precisely the strict coercivity on V of the bilinear form a(, ) introduced in Equations (5) and (7)). We prove below this strict coercive property. We assume the functions to be real. First, we need to show that the bilinear form a(, ) is continuous, that is, that a(u, v) C u K v K. Since ρ = r in the neighborhood V i of Q i S and RV is continuous on V i, we have V Cρ 2. Therefore, the Cauchy-Schwarz inequality gives a(u, v) C u K v K, for C > 0 not depending on u or v. We prove strict coercivity of a(, ) on each V i and on Ω O, O = V i /2, respectively. Assume D Ω is not empty, then, on the later set, our desired inequality is just the usual Poincaré inequality. The case when V 0 but is not identically zero is similar. Then, we shall verify the following inequality on every V i. V i [ ( x u) 2 + ( y u) 2 + V u 2] dxdy C u 2 K (Vi). Assume first that Q i is a vertex with angle α i (0, 2π) and at least one side of Q i has Dirichlet boundary conditions. Then, V i can be locally characterized in polar coordinates with Q i at the origin by V i = {(r, θ), 0 < r < l, 0 < θ < α i }. Note that a(u, u) Ω [( xu) 2 + ( y u) 2 ]dxdy. Therefore, it suffices to show Vi u 2 dxdy Ca(u, u), C > 0, r2 because r = ρ on V i. Assume that the positive x-semiaxis corresponds to the side with Dirichlet boundary conditions (this is where we need the assumption that there are no adjacent Neumann sides). The one-dimensional Poincaré inequality for θ on V i then gives, for each fixed r, αi αi u 2 dθ C ( θ u) 2 dθ. By integrating in polar coordinates, we have Vi u 2 0 r 2 dxdy = Vi u 2 0 r drdθ C ( θ u) Vi 2 drdθ. r Since V i [( x u) 2 + ( y u) 2 ]dxdy = V i [r( r u) 2 + ( θ u) 2 /r]drdθ, we get u Vi 2 ρ Vi 2 dxdy = u 2 r 2 dxdy C [ ( x u) 2 + ( y u) 2 + V u 2] dxdy. V i

12 2 H. LI AND V. NISTOR On the other hand, if Q i S Ω (that is, if Q i is an interior point) or if Q i is a point separating two adjacent sides endowed with Neumann boundary conditions, we have V δr 2, δ > 0 (this is where we use the assumption RV (Q i ) > 0). We can no longer use the one dimensional Poincaré inequality, but do not need it either, because we can write [ u 2 + V u 2] [ dxdy δ u 2 + u 2 /r 2] dxdy =: δ u 2 K (Vi), V i V i for δ > 0 small enough (after we decrease the neighborhoods V i, if necessary). The strict coercivity of H (or of the bilinear form a) on V follows by adding all these inequalities. The Lax-Milgram Lemma then proves that H : V V is an isomorphism, which is our result for m = 0 and a = 0. We next use the continuity of the family ρ ɛ Hρ ɛ : V V and Lemma 2.5 to prove the result for m = 0 and a < η, for some η that depends only on the domain and the operator H. Theorem 3. shows that if the result is true for (0, a), then it is true also for (m, a). This completes the proof An application to the Finite Element Method. The proof of our wellposedness result leads to an important application to the Finite Element approximations of the solution u or the mixed Boundary Value Problem (6). We shall assume therefore that in this subsections the assumptions of Theorem 3.3 are satisfied. Recall the bilinear form a(u, v) = ( u, v) + (V u, v) first defined in the Introduction, Equation (5), where the inner products are the inner products in the real L 2 (Ω). Also, recall the main variational space V := K (Ω) {u = 0 on DΩ}, also first defined in the Introduction, in Equation (4). The proof of Theorem 3.3 then gives the following corollary. Corollary 3.4. Let us denote by u a := a(u, u) /2 the energy norm defined by a. Assume the conditions of Theorem 3.3 are satisfied. Then the energy norm a and the defining norm of K := K (Ω) are equivalent on V. Proof. This is nothing but another way of stating that the assumptions of the Lax- Milgram theorem are satisfied by the bilinear form a on V V, which we checked in the proof of Theorem 3.3 Remark 3.5. In case Ω S = and there are no Neumann/Neumann corners, then the energy norm a is also equivalent to the H semi-norm χ H := χ L 2. This is well known (see, for example, [3] and the references therein). Let us fix a subspace S V. In our applications, S will be a finite dimensional subspace defined using a typical Finite Element procedure starting from a mesh (or triangulation) of our domain. This is however not necessary for the following Ceatype Lemma. Recall that u S S denotes the discrete solution of (6). Given suitable f and g N, the discrete solution u S S is defined by (8) a(u S, v S ) = (f, v S ) L 2 (Ω) + g N, v S NΩ, v S S. Proposition 3.6. Assume the conditions of Theorem 3.3 are satisfied, then there exists a constant C > 0, independent of f and g N, such that u u S K C inf χ S u χ K.

13 MODIFIED SCHRÖDINGER OPERATOR AND FEM 3 Proof. The proof is standard. Indeed, we have u u S a = inf χ S u χ a, because u S is the projection of u onto S in the a-inner product. The result then follows from the equivalence of the a-norm (i. e., energy norm) and the K -norm on V (corollary 3.4). Estimates on inf χ S u χ K, for suitable χ, will be obtained in the following section, Section 4. These estimates depend, to a certain extend, on η, so we now provide a determination of the optimal value of η in Theorem Determination of η. For our constructions, it may be necessary to have estimates on η. In fact, it turns out that in two dimensions it is possible to explicitly determine the optimal value of η in Theorem 3.3. To this end, let us define (9) δ i = lim Q Q i Q Q i 2 V (Q) for any Q i S. By freezing the coefficients of H to Q i, we see that the behavior of the solution u on V i is controlled by + δ i /r 2 or, more precisely, by its operator pencil s spectrum Σ i [34, 35], which we proceed to describe. The operator pencil P i (τ) (or indicial family) associated to H at Q i is defined by (20) ( + δ i /r 2 )(r ıτ+ɛ φ(θ)) = r ıτ+ɛ 2 P(τ)φ(θ), which gives (2) P i (τ) := (τ ıɛ) 2 2 θ + δ i. We shall distinguish four cases, the case when Q i is an interior point and the three cases corresponding to Q i on the boundary and to the three possible choices for boundary conditions on the two sides abutting to Q i : Dirichlet/Dirichlet (D/D) boundary conditions, Dirichlet/Neumann (D/N) boundary conditions, and Neumann/Neumann (N/N) boundary conditions. Let us denote by α i the angle of Ω at Q i (α i = 2π if Q i is an interior point). Denote 2N = {2, 4, 6,...}, N /2 = {/2, 3/2,...}, as usual. We then define Σ i = {± λ} with λ an eigenvalue of θ 2+δ i on [0, α i ] with suitable boundary conditions (periodic, D/D, D/N, or N/N). This gives { } (22) Σ i := ± k 2 π 2 α 2 i + δ i where k satisfies 2N {0} if Q i is an interior point N if D/D boundary conditions are assigned at Q i k N /2 if D/N boundary conditions are assigned at Q i N {0} if N/N boundary conditions are assigned at Q i. Then P(τ) is invertible for all τ R, as long as ɛ Σ i. For these values of ɛ, we can use Kondratiev s method to obtain Fredholm conditions on ρ ɛ Hρ ɛ. See [38, 58] and the references therein for the case of interior singularities due to the potential V, where no boundary is involved (in which case, a suitable pseudodifferential calculus immediately gives the desired Fredholm condition.) See also Theorem 4. in Schwab s book [59] for the relation with singular functions in the case when Q is on the boundary.

14 4 H. LI AND V. NISTOR Recall that a continuous operator A : X Y between Banach spaces is Fredholm if the kernel of A (that is the space ker(a) := {Ax = 0}) and Y/AX are finite dimensional spaces (we shall not consider unbounded Fredholm operators in this paper). If A : X Y is a continuous Fredholm operator, we defined its index by the formula ind(a) = dim ker(a) dim(y/ax). Theorem 3.7. Assume that δ i 0 and that the General Assumptions are satisfied. Then the mixed Boundary Value Problem (6) defines a Fredholm operator (23) Ha := (H, res, ν ) : K a+ (Ω) Km a (Ω) K/2 a+/2 ( DΩ) K m /2 a /2 ( NΩ) for all a i Σ i, with res being the restriction operator to D Ω. We are now ready to determine the value of η in Theorem 3.3. Let η 0 be the minimum values of a, a i Σ i. Then η 0 > 0 exactly if the additional conditions of Theorem 3.3 are satisfied. Let us therefore assume for the rest of this subsection that η 0 > 0. Then we have the following. Proposition 3.8. Assume that η 0 > 0. Then the assumptions of Theorem 3.3 are satisfied and η = η 0 is the largest value for which that theorem is valid. Proof. We shall use the notation and the ideas of the proof of Theorem 3.3. As in the proof of that theorem, we can assume m = 0. Let H a = ρ a Hρ a. The operator H a is invertible precisely when H a is invertible (because we have assumed m = 0 and the trace map is continuous and surjective). It is enough then to show that H a is invertible if a < η 0, since H a is not Fredholm for a = η 0. Let us therefore assume that a < η 0. Then H a is Fredholm, by Theorem 3.7. Theorem 3.3 states that H a is invertible for a = 0. By the homotopy invariance of the index, Ha is Fredholm of index zero for a < η 0. Let us notice next that the domain ρ a V of the operator H a decreases as a increases. Hence, the kernel of H a decreases as a increases. We therefore conclude that H a is injective for 0 a < η. Since it has index zero, it is in fact an isomorphism. Since Ha = H a, we obtain the invertibility of H a for η < a 0 as well. This completes the proof The case η 0 = 0. Although we shall not need it in this paper, let us quickly take a look at the case when we do not have invertibility in Theorem 3.3, namely when η 0 = 0. The operator H a will no longer be invertible for a η. In that case, even if a is away from the specific values above, Ha is only a Fredholm operator with a non-zero index that can be computed using the results of [34, 35, 48, 5]. The result is as follows. Let us fix a < a and count the number of values in the set (a, a ) Σ i. Due to multiplicities, we count twice the values corresponding to one of the following situations: (a) Q i is an internal point and k 0, (b) Q i is an internal point k = 0 and δ i = 0, (c) Q i is a Neumann/Neumann point, k = 0, and δ i = 0. Let N be the total number obtained from this counting procedure. Assume that a, a Σ i. Then (24) ind( H a ) ind( H a ) = N. The same result applies to the operator H a : ρ a B ρ a B, which satisfies H a = H a. This allows to determine the index of H a. This amounts to an index problem

15 MODIFIED SCHRÖDINGER OPERATOR AND FEM 5 both for the Neumann and for the Schrödinger problems in the plane. Recall that V is real valued. Theorem 3.9. Assume the conditions of Theorem 3. are satisfied and let N be the number of points Q i S that do not satisfy the conditions of Theorem 3.3 (more precisely δ i = RV (Q i ) = 0 at an interior point or at a Neumann-Neumann point). Then H a is Fredholm for a < η, a 0, with index ind( H ±a ) = N, 0 < a < η. 4. Estimates for the Finite Element Method From now on we shall assume that our domain Ω coincides with a straight polygonal domain P, that is, with a domain whose boundary is piecewise linear. We fix the degree of approximation m. We shall also assume that the operator H satisfies the assumptions of Theorem 3.3. This means that V 0, that V R(Q) > 0 at the interior points of S and at the Neumann/Neumann vertices, and that the General Assumptions introduced in the previous section are satisfied. Our goal is to apply the Finite Element method to approximate the solution u of our Schrödinger type mixed boundary value problem (2) (this is the same equation as the mixed Boundary Value Problem (6) of the previous section with g D = 0 and g N = 0, that is, with zero boundary data). Let m N be our fixed constant. In this section, we construct a sequence of meshes T n with S n the associated Finite Element spaces of piecewise polynomials of degree m and prove that T n yields quasi-optimal rates of convergence for the sequence of finite element approximations u n := u Sn of u. Namely, u u n K C dim(s n ) m/2 f H m, f H m. The proof of this estimate is part of our main theorem, Theorem 4.. Quasi-optimal rates of convergence for elliptic problems with smooth coefficients on polygonal domains were obtained before in [4,, 3, 26, 47, 56] using adaptive meshes (usually graded meshes, like in this paper). We shall thus concentrate our attention on the refinement of our meshes close to the interior singularities of V. Around the other points of S, we shall use the grading explained in [3, 4]. 4.. Approximation away from the singular set S. In this subsection, we approximate the solution u far from the singular points, so we ignore the role of the singular set S. In particular, although the results and constructions of this subsection are formulated for P, often they will be used for a subpolygon G P. We first need to recall the following well-known approximation theorem [9, 2, 23, 59]. Let T = {T } be a mesh, that is a triangulation of P with triangles T. We shall denote by S(T, ) the Finite Element space associated to the degree m Lagrange triangle. That is, S(T, m + ) consists of all continuous functions χ : Ω R such that χ coincides with a polynomial of degree m on each triangle T T. (The smaller subspace S(T, ) := S(T, ) V will be used in our approximation results in the following subsections. It is more convenient in this subsection to use the larger subspace S(T, ), though. This also allows for more general results.) We shall denote by u I = u I,T, S(T, m + ) the Lagrange interpolant of u H 2 (P). Let us recall the definition of u I,T, for the benefit of the reader. First, given a triangle T, let [t 0, t, t 2 ] be the barycentric coordinates on T. The nodes of the degree m Lagrange triangle T are the points of T whose barycentric

16 6 H. LI AND V. NISTOR coordinates [t 0, t, t 2 ] satisfy mt j Z. The degree m Lagrange interpolant u I,T, of u is the unique function u I,T, S(T, m + ) such that u = u I,T, at the nodes of each triangle T T. The shorter notation u I will be used only when only one mesh is understood in the discussion (recall that m is fixed). Theorem 4.. Let T be a triangulation of P. Assume that all triangles T of the triangulation T on the domain D have angles α and edges of length h. Let u H (P) and let u I := u I,T, S(T, m + ) be the degree m Lagrange interpolant of u. Then, there exist a constants C(α, m) > 0 independent of u such that u u I H (P) C(α, m)h m u H (P). Let M := C(α, m + )M M 2, where C(α, m + ) is as in Theorem 4. and M and M 2 are as in Lemma 2.7. Then, we have the following estimate for the error u u I K (G) on a subset G P that is away from any point Q i S. Recall that ρ is the function defined by Equation (0). Proposition 4.2. Fix α > 0 and 0 < ξ < l. Let G P be a polygonal subset, such that ρ > ξ on G. Let T = (T j ) be a triangulation of G with angles α and sides h. Then u u I K (G) Mh m u K a+ (G) for all u Ka+ (G), where M depends only on ξ, α, and m. Proof. The proof of this result is obtained from Theorem 4. and the equivalence of the H m -norm and the Ka m -norm on G (Lemma 2.7). We now want to extend the above results close to the vertices. While the final estimates will be given in the next subsection, we include here a few more intermediate results. Let us fix for the rest of this subsection a point Q i S. We shall need to study the local behavior with respect to dilations of a function v K m a = K m a (P) with support in the neighborhood V i of Q i S. Therefore, for the sake of simplicity, we shall consider a new coordinate system that is a simple translation of the old xy-coordinate system with Q i now at the origin of the new coordinate system. Let G V i be the subset, such that ρ ξ l on G. For 0 < λ <. We let G := λg. Then, we define the dilation of a function on G in the new coordinate system as follows, v λ (x, y) := v(λx, λy) for all (x, y) G V i. (This definition makes sense, since Q i is the origin in the new coordinate system.) We shall need the following dilation lemma. Lemma 4.3. Let 0 < λ < and G V i be an open subset such that G := λg V i. Then u λ K m a (G) = λ a u K m a (G ) for any u K m a (V i ). Proof. The proof is based on the change of variables w = λx, z = λy. Note that on both G V i and G V i, ρ(x, y) is equal to the distance from (x, y) to Q i, hence

17 MODIFIED SCHRÖDINGER OPERATOR AND FEM 7 ρ(x, y) = λ ρ(w, z). Then, u λ (x, y) 2 K m a (G) = ρ j+k a (x, y) x j k y u λ(x, y) 2 dxdy j+k m G = λ a j k ρ j+k a (w, z)λ j+k w j k z u(w, z) 2 λ 2 dwdz j+k m G = λ 2a 2 ρ j+k a (w, z) w j zu(w, k z) 2 dwdz j+k m G = λ 2a 2 ρ j+k a (w, z) w j zu(w, k z)) 2 L 2 (G ) j+k m = λ 2a 2 u 2 K m a (G ), which completes the proof. We continue to fix the point Q i S. Let V i := P B(Q i, l), be as in Equation (9) and T ξ V i be a triangle with the biggest edge of length = ξ. We assume that Q i is a vertex of T ξ. Denote by T κξ T ξ the sub-triangle of T ξ that has Q i as a vertex as well, is similar with T ξ with the ratio of similarity κ, 0 < κ < and has all sides parallel to the sides of T ξ. Then, T ξ is divided into the small triangle T κξ that has the common vertex Q i with T ξ, and the trapezoid between the two parallel edges. We then have the following estimate near Q i. Proposition 4.4. Let 0 < κ <, α > 0 and T κξ T ξ V i be the triangles described above. Let T = (T j ) be a triangulation of G := T ξ T κξ with angles α and edges h. Then the degree m Lagrange interpolant u I,T satisfies u u I,T K (G ) C(κ, α, m)ξ a (h/ξ) m u K a+ (G ) for all u K m a+ (V i), a 0, with C(κ, α, m) independent of ξ, h, and u. Proof. We use Lemma 4.3 with λ = ξ/ l. Recall the dilation function u λ (x, y) = u(λx, λy), and note that dilation commutes with interpolation in the sense that u Iλ = u λi, with the first interpolant defined using the mesh T and the second interpolant defined using the dilation of this mesh by λ. Thus, we can apply Proposition 4.2 to the region G = λ G V i, u u I K (G ) = u λ u Iλ K (G) = u λ u λi K (G) where the last inequality is from Lemma 2.6. M(h/λ) m u λ K (G) = M(h l/ξ) m u K (G ) C(κ, α, m)ξ a (h/ξ) m u K a+ (G ), 4.2. Construction of the finite element spaces. Recall that m is fixed and that V := K (Ω) {u = 0 on D Ω}. For any mesh T of P, we let (25) S(T, m + ) := S(T, m + ) V = {χ S(T, m + ), χ = 0 on D P S}.

18 8 H. LI AND V. NISTOR The condition χ = 0 on D P in the above equation is due to the fact that our main variational space V consists of functions that vanish on D P. The condition χ = 0 at S is due to the fact that a(χ, χ) < for all χ V. Recall that we want to construct a sequence of meshes T n, with Finite Element spaces S n := S(T n, m + ), such that the sequence u n := u Sn S n of Galerkin approximations of the solution u for our Schrödinger type mixed boundary value problem, Equation (2), satisfies u u n K C dim(s n ) m/2 f H m. We shall achieve this quasi-optimal rate of convergence by considering a suitable grading close to the points of S. The proof will be based on estimating the error in weighted Sobolev spaces. At the boundary, the estimates are well known [4,, 3, 56], so we shall now concentrate at an interior point Q i where V is singular. The meshes T n that are used to define the spaces S n = S(T n, m + ) will be constructed by successive refinements and will have the same number of triangles as the meshes obtained by the usual mid-point refinement. Let η = min Σ j, which satisfies Theorem 3.3, by Proposition 3.8. From now on, we shall assume that the right hand side f of Equation (2) satisfies the condition f H m := H m (P) Ka m := Km a (P), where 0 a < min(η, ), m. Therefore the solution u of Equation (2) satisfies u K a+ {u DP = 0} V := K (Ω) {u = 0 on D Ω}, by Theorem 3.3. We now introduce our refinement procedure. Recall that the vertices of P and the points where the boundary conditions change are contained in S. Definition 4.5. Let κ (0, /2] and T be a triangulation of P such that no two vertices of P belong to the same triangle of T. Then the κ refinement of T, denoted κ(t ) is obtained by dividing each edge AB of T in two parts as follows. If neither A nor B is in S, then we divide AB into two equal parts. Otherwise, if say A is in S, we divide AB into AC and CB such that AC = κ AB. This will divide each triangle of T into four triangles. We now introduce our sequence of meshes. Recall that l > 0 was introduced in Remark 2. and 4 l is not greater than the distance from a point Q in S to an edge of P that does not contain it. Definition 4.6. For a fixed m = {, 2,...}, we define a sequence of meshes T n as follows. The initial mesh T 0 is such that each edge in the mesh has length l/2 and every point in S has to be the vertex of a triangle in the mesh. In addition, we chose T 0 such that there is no triangle in T 0 that contains more than one point in S. Then we define by induction T n+ = κ(t n ) (see Definition 4.5). We shall denote by u I,n = u I,Tn, S n := S(T n, m + ) the degree m Lagrange interpolant associated to u C(P) and the mesh T n on P. Note that near the vertices, our refinement coincides with the ones introduced in [4,, 3, 56]. We now investigate the approximation properties afforded by the triangulation T n close to a fixed point Q i S. We also fix a triangle T T 0 that has Q i as a vertex. Let us denote by T κ j = κ j T T the small triangle belonging to T j that is similar to T with ratio κ j, has Q j as a vertex, and has all sides parallel to the sides

19 MODIFIED SCHRÖDINGER OPERATOR AND FEM 9 Figure. One refinement of triangle T with vertex Q, κ = l /l 2 of T. Then T κ j T κ j. Moreover, since κ < /2 and the diameter of T is l/2, we have T κ j V i, j, by the definition of V i. Let N be the level of refinement. In all the statements below, h 2 N, in the sense that they have comparable magnitudes. In particular, we can replace h with 2 N in all the estimates below, possibly by increasing the constants. A good choice is h = h 0 2 N, where h 0 is the initial mesh size. We shall need the following general lemma. Lemma 4.7. We have K 2 := K2 (P) L (P) and K+a 2 C(P), for a > 0, and hence every function u K+a 2 is continuous and vanishes on S. Proof. Let G = T κ j T κ j. The Sobolev embedding theorem gives u L (G) C u K 2 (G). The constant can be chosen to be independent of j since both norms are dilation invariant (this is obvious for the L norm and for the K 2 norm it follows from Lemma 4.3). This shows that K 2 L (P). The classical Sobolev embedding theorem also shows right away that K 2 C(P S). The relation K+a 2 = ρ a K 2 and the fact that ρ a (p) 0 as p Q S shows that any u K+a 2 is also continuous at every point of S. It follows from the above lemma that the interpolant u I is defined for u K +a and u I (Q) = 0 for all Q S. See [] for more general embedding theorems. Lemma 4.8. Let 0 < κ 2 m/a, 0 < a < η. Let us consider the small triangle T κ N = κ N T T with vertex Q i, obtained after N refinements. Let u I,N be the degree m Lagrange interpolant of u associated to T N. Then, on T κ N T N, we have u u I,N K (T κ N ) Ch m u K a+ (T κ N ), for all u K a+ (V i) {u DP = 0}, h /2 N, where C depends on m and κ, but not on N. Proof. Let us denote u λ (x, y) = u(λx, λy) with Q i as the origin. Let λ = κ N. Then, u λ (x, y) Ka+ (T) by Lemma 4.3. Let χ : G [0, ] be a non-decreasing smooth function that depends only on ρ and is equal to 0 in a neighborhood of Q i, but is equal to at all the nodal points different from Q i. We introduce the auxiliary function v = χu λ on T. Consequently, v 2 K (T) = χu λ 2 K (T) C u λ 2 K (T), where C depends on m and the choice of the nodal points. Moreover, since u(q i ) = 0 by Lemma 4.7, the interpolant v I = u λi = u Iλ on T by the definition of v.

20 20 H. LI AND V. NISTOR This gives u u I K (T κ N ) = u λ v + v u λi K (T) u λ v K (T) + v u λi K (T) = u λ v K (T) + v v I K (T) C u λ K (T) + C 2 v K (T) C u λ K (T) + C 3 u λ K (T) = C u K (T κ N ) + C 3 u K (T κ N ) C 4 κ Na u K a+ (T κ N ) Ch m u K a+ (T κ N ).. The first and the sixth relations above are due to Lemma 4.3; the fourth is due to Proposition 4.2; and the seventh is based on Lemma 2.6. We now combine the estimate on T κ N of the previous lemma with the estimates on the sets of the form T κ j T κ j+ of Proposition 4.4 to obtain the following estimate on an arbitrary, but fixed, triangle T T 0 that has a vertex in S (the difficult case not handled by Proposition 4.2). We continue to fix a triangle T of T 0 with a vertex Q i S. Proposition 4.9. Denote by h /2 N the mesh size of T N and let 0 < κ 2 m/a, 0 < a < η. Then there exists a constant C > 0, such that for all K a+ (P). u u I,N K (T) Ch m u K a+ (T), Proof. The proof follows from the estimates on the subsets T κ j T κ j, j N, (Proposition 4.4) and from the estimate on T κ N (Lemma 4.8). Definition 4.4 shows that the mesh size of T κ j T κ j, is κ j 2 j N. Then, using the notation in Proposition 4.4, we have ξ = O(κ j ), therefore, u u I K (T κ j T κ j ) C κ (j )a (κ j 2 j N /κ j ) m u K a+ (T κ j T κ j) C 2 2 (j )m 2 Nm+(j )m u K a+ (T κ j T κ j ) = C 2 2 Nm u K a+ (T κ j T κ j) Ch m u K a+ (T κ j T κ j), where C depends on κ, but not on the subset T κ j T κ j. Since the estimate of the interpolation error on T κ N has been given in Lemma 4.8, we complete the proof of Proposition 4.9 by adding up the error estimates on all the subsets T κ j T κ j, j N, and on T κ N. Remark 4.0. Denote by T be the union of all the initial triangles that contain singular points of V. Then T is a neighborhood of S in P. Moreover, the interpolation error on T also satisfies u u I K (T) Ch m u K a+ (T) by summing up the estimates in Proposition 4.9 over all the triangles, as long as κ is chosen appropriately. Here we state our main result, namely the quasi-optimal convergence rate of the numerical solutions on our meshes.

21 MODIFIED SCHRÖDINGER OPERATOR AND FEM 2 Theorem 4.. Let 0 < a < η and 0 < κ < 2 m/a, with m fixed. We assume that the conditions of Theorem 3.3 are satisfied. Let T n be obtained from the initial triangulation by n refinements, as in Definition 4.6. Let S n := S n (T n, m + ) be the Finite Element space given by Equation (25) and u n = u Sn S n be the Finite Element solution defined by Equation (6). Then there exists C > 0 such that for any f K m a. u u n K Ch m f K m, h 2 n. a Proof. Let T i be the union of initial triangles that contain Q i S. Recall from Theorem 3.3 that u K C f a+ K m. We use the previous estimates to obtain a u u n K C u u I K = C ( u u I K (P T i) + ) u u I K (T i) Ch m( u K (P T i ) + u K a+ (Ti)) Ch m u K a+ Ch m f K m. a The first inequality is based on Céa s Lemma (Proposition 3.6) and the second inequality is based on the Propositions 4.2 and 4.9. Then, as a direct result of the theorem above, we have the following estimate on the convergence rate of the finite element solution, which indicates that it is quasi-optimal. Theorem 4.2. Using the notation and assumptions of Theorem 4., we have that u n = u Sn S n := S(T n, m + ) satisfies for a constant independent of f and n. u u n K C dim(s n ) m/2 f K m, a Proof. Let T n be the triangulation of P after n refinements. Then, the number of triangles is O(4 n ) based on the construction of triangles in different levels. Therefore, the dimension of S, dim(s n ) 4 n, for Lagrange triangles. Thus, from Theorem 4., the following estimates are obtained, u u n K C h m f K m a The proof is complete. C2 nm f K m a C dim(s n ) m/2 f K m. a Using that H m Ka m for a (0, ), we obtain the following corollary, under the assumptions of the above theorem. Corollary 4.3. Let 0 < a min{, η} and 0 < κ < 2 m/a. Then for a constant independent of f and n. u u n K C dim(s n ) m/2 f H m,

22 22 H. LI AND V. NISTOR 5. Numerical Results We shall present here some numerical results that will illustrate the effectiveness of our mesh refinement techniques used for numerically solving our Schrödinger type mixed boundary value problem (2). Our numerical tests will convincingly show that our sequence of meshes achieves quasi-optimal rates of convergence. As we have seen in Section 3, the singularity of the solution near Q i S heavily depends on the parameter δ i := lim p Q i V (p)r(p), and the behavior of the solution u near a singular point Q i S is local, in the sense that it depends only on local information at Q i. Therefore, we shall assume in this section that H is of the form H = + δr 2 as model problems, where r is the distance to the origin O = Q 0. We shall consider different values of δ > 0, as well as different positions of the origin on the domain, in order to verify more thoroughly our theoretical results. Note, however, that in the model problems below, we shall chose m =, namely, we shall chose piecewise linear functions for the definition of our Finite Element spaces S(T, ) introduced in Equation (25). This will make the implementation simpler, while the results will still be quite relevant. A crucial role in our implementation is played by the parameter κ introduced in the previous section. This is the parameter that controls the way we perform the mesh refinement and thus, together with the initial refinement, completely determines our sequence of meshes. We shall see that a choice of κ in the acceptable range (0, 2 m/η ) yields quasi-optimal rates of convergence, whereas a choice of κ away from this range will not give quasi-optimal rates of convergence. Since m = in our implementations, the threshold value is 2 /η. Note that κ decreases as η decreases, which explains why we want a good determination of η: a smaller η results in triangles with smaller angles. Intuitively, this seems to be a bad thing, but in practice we have not seen any inconvenience. Our analysis of this issue is far from complete, though. 5.. Interior singularity. Let us first consider the following two model problems on the square domain ˆP := (, ) (, ) { u + 0.5r 2 u = in (26) ˆP, u = 0 on ˆP, (27) { u + 2r 2 u = u = 0 in ˆP, on ˆP. (So the origin Q 0 is an interior point of the domain in these examples.) Our results show that for the problem (26), the solution is not in H 2 near the origin. Therefore, a special refinement of the mesh is needed near the origin to ensure quasi-optimal rates of convergence in the Finite Element method. To be more precise, from our theory developed above (Equation (22)), we can take a value of a, such that 0 < a < η = , which makes κ = 2 /a < 2 /η

23 MODIFIED SCHRÖDINGER OPERATOR AND FEM 23 Figure 2. Initial triangles for ˆP (left); the mesh after four refinements, κ = 0.2 (right) j\κ e : 0. e : 0.2 e : 0.3 e : 0.4 e : 0.5 e : 0. e : 0.2 e : 0.3 e : 0.4 e : Convergence history for (26) Convergence history for (27) e = log 2 ( u j u j H u j+ u j H ). Table. Interior singularities In fact, a more accurate a prior estimate [34] on the solution gives u H s for s < The situation for the problem (27) is different, a consequence of the the regularity of the solution depends on the parameter δ. Based on our method, η = 2.44 >, which means the solution is in H 2, and hence no graded mesh is necessary when using piecewise linear approximating functions. Figure 2 shows the mesh refinement for some small values of n. Note that we can use uniform meshes near the corners of the domain since they correspond to angles < π. Table lists the convergence rates of the finite element solutions for equations (26) and (27), respectively, on triangulations with different values of κ. These results verify our theoretical prediction: the quasi-optimal convergence rates can be obtained as κ < for Equation (26); no graded meshes are needed to approximate the solution of Equation (27) to get the quasi-optimal rates. The left most column in the table shows the number of the refinement levels, and u j represents the numerical solution on the mesh after j refinements. The quantities printed out in other columns in the table are the convergence rates defined by e = log 2 ( u j u j H ), u j+ u j H