discrete maximum principle for interior penalty discontinuous galerkin methods

Cent Eur J Math • 11(4) • 2013 • 664-679 DOI: 10.2478/s11533-012-0154-z ❈❡♥tr❛❧ ❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s Discrete maximum principle for interior penalty discontinuous Galerkin methods ❘❡s❡❛r❝❤ ❆rt✐❝❧❡ Tamás L Horváth1,2∗ , Miklós E Mincsovics1,2† Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány P sétány 1/C, Budapest, 1117, Hungary MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Pázmány P sétány 1/C, Budapest, 1117, Hungary ❘❡❝❡✐✈❡❞ ✷✼ ❋❡❜r✉❛r② ✷✵✶✷❀ ❛❝❝❡♣t❡❞ ✷✾ ❆♣r✐❧ ✷✵✶✷ ❆❜str❛❝t✿ A class of linear elliptic operators has an important qualitative property, the so-called maximum principle In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter We then investigate the sharpness of these conditions The theoretical results are illustrated with numerical examples ▼❙❈✿ 65N30, 35B50 ❑❡②✇♦r❞s✿ Discrete maximum principle • Discontinuous Galerkin • Interior penalty © Versita Sp z o.o Introduction When choosing a numerical method to approximate the solution of a continuous mathematical problem we need to consider which method results in an approximation that is not only close to the solution of the original problem, but also shares the important qualitative properties of the original problem For linear elliptic problems the most important qualitative property is the maximum principle The reader can find detailed explanations from different viewpoints about the importance of the preservation of the maximum principle in [9, Sections 1,2], [13, Section 1], and [5] ∗ † ✻✻✹ E-mail: thorvath@cs.elte.hu E-mail: mincso@cs.elte.hu T.L Horváth, M.E Mincsovics The preservation of the maximum principle was extensively investigated for finite difference methods (FDM) and for finite element methods (FEM) with linear and continuous elements, but not in the context of the discontinuous Galerkin method In this paper we take the first step to fill this gap Namely, we investigate an interior penalty discontinuous Galerkin method (IPDG) applied to a 1D elliptic operator (containing diffusion and reaction terms) and we show that it is possible to give reasonable and sufficient conditions for the maximum principle on the discrete level The paper is organized as follows In Sections and we give a short overview on continuous and discrete maximum principles including the important notions and preliminary results that we will use later on In Section we deal with the IPDG method applied to some 1D elliptic operator In Section we give conditions under which the discrete maximum principle holds In subsection 6.1 we investigate the sharpness of our conditions with the help of numerical examples We conclude the investigation in subsection 6.2 We include an appendix about the Z- and M-matrices for the readers’ convenience since we use these notions throughout Section Continuous maximum principle for elliptic operators We define the maximum principle for operators, following the book [8], instead of defining it for equations Naturally, there are no important differences between the two approaches, but our choice is easier to handle Let Ω ⊂ Rd be an open and bounded domain with boundary ∂Ω, and Ω = Ω ∪ ∂Ω its closure We investigate the elliptic operator K , dom K = H (Ω), defined in divergence form as d Ku = − ∂ ∂x j i,j=1 aij d ∂u ∂xi bi + i=1 ∂u + cu, ∂xi (1) where aij (x) ∈ C (Ω), bi (x), c(x) ∈ C (Ω) Note that smoothness of the coefficient functions gives the opportunity to rewrite (1) to a non-divergence form that is more suitable for the investigation of maximum principles Definition 2.1 We say that the operator K , defined in (1), possesses the (continuous) maximum principle if for all u ∈ C (Ω) ∩ C (Ω) the following implication holds Ku ≤ in Ω =⇒ max u ≤ max 0, max u ∂Ω Ω Theorem 2.2 ([8, Chapter 6.4, Theorem 2]) If operator K , defined in (1), is uniformly elliptic and c ≥ 0, then it has the continuous maximum principle Maximum principle for FEM elliptic operators – short overview 3.1 The construction of the FEM elliptic operator When discretizing the operator (1) with some finite element method we have to define the corresponding bilinear form d a(u, v) = aij Ω i,j=1 ∂u ∂v + ∂xi ∂xj d bi i=1 ∂u v + cuv dx, ∂xi (2) where u ∈ H (Ω), v ∈ H01 (Ω) We note that this means we deal with nonhomogeneous Dirichlet boundary condition, for the homogeneous one see Remark 3.4 ✻✻✺ Discrete maximum principle for interior penalty discontinuous Galerkin methods The following step is to define a mesh on Ω A 1D mesh consists of intervals The discrete maximum principle literature focuses on regular triangle or hybrid meshes (containing both triangles and rectangles) in 2D and tetrahedron or block meshes in 3D A given mesh determines the sets P = {x1 , x2 , , xN } and P∂ = {xN+1 , xN+2 , , xN+N∂ } containing the vertices in Ω and on ∂Ω, respectively Let us introduce two more notations: N = N + N∂ and P = P ∪ P∂ Next we can define a subspace of H (Ω) corresponding to the mesh This can be done by giving a basis of this subspace The basis functions are denoted by Φi (x), i = 1, , N The discrete maximum principle literature investigates almost solely the case of hat-functions which are defined with the following properties: the basis functions are continuous; the basis functions are piecewise linear over triangles/tetrahedrons and multilinear over rectangles/blocks; Φi (xi ) = for i = 1, , N; Φi (xj ) = for i, j = 1, , N, i = j Note that due to such choice, the subspace consists of continuous functions; N i=1 φi (x) = holds for all x ∈ Ω; Φi (x) ≥ holds for all x ∈ Ω and i = 1, , N; in a linear combination of the basis functions the coefficients represent the values of the resulting function at the points of P (We remark that for higher order elements the investigation is more difficult, and positive results are obtained only for a simple 1D problem, see [17]; for a higher dimensional case, in [12] negative results are obtained.) Finally, we can construct the so-called stiffness matrix K ∈ RN×N as Kij = a(Φj , Φi ), that is, the discrete operator corresponding to (1) In the following it will be useful to introduce the partitioned form K = [K0 |K∂ ], where K0 ∈ RN×N , K∂ ∈ RN×N∂ , acting on the vector u = [u0 |u∂ ]T ∈ RN , u0 ∈ RN , u∂ ∈ RN∂ , which is constructed by taking into consideration the separation of the (discrete) interior and boundary nodes 3.2 Maximum principle for FEM elliptic operators To define the corresponding discrete maximum principle we introduce some notation The symbol denotes the zero matrix (or vector), e is the vector all coordinates of which are equal to The dimensions of these vectors and matrices should be clear from the context Inequalities A ≥ or a ≥ mean that all elements of A or a are nonnegative By max a we denote the maximal coordinate of the vector a Now we are ready to define the corresponding discrete maximum principle for the matrix K Definition 3.1 ([4]) We say that a matrix K has the discrete maximum principle if the following implication holds: Ku ≤ =⇒ max u ≤ max {0, max u∂ } Note that this definition is adequate only because the chosen basis functions have special properties E.g., for higher order basis functions this definition is not applicable It is relatively easy to give sufficient and necessary conditions for this principle ✻✻✻ T.L Horváth, M.E Mincsovics Theorem 3.2 ([4]) The matrix K possesses the discrete maximum principle if and only if the following three conditions hold: (T1) K0−1 ≥ 0, (T2) −K0−1 K∂ ≥ 0, (T3) −K0−1 K∂ e ≤ e Theorem 3.2 is a theoretical result and difficult to apply directly Usually these conditions are relaxed with the following practical conditions Theorem 3.3 ([4]) The matrix K has the discrete maximum principle if the following three conditions hold: (P1) K0 is a nonsingular M-matrix, (P2) −K∂ ≥ 0, (P3) Ke ≥ For the definition of M-matrix see Definition 7.2 The reader can find detailed information and a plentiful reference list about the discrete maximum principle in [16] For attempts to use less restrictive practical conditions we recommend the papers [15] and [10] Remark 3.4 Note that if we apply the homogeneous Dirichlet boundary condition  this is the case when we eliminate the boundary condition at the continuous level  then the matrix K∂ has no effect, which results in that we need to guarantee (T1) or (P1) only This milder property has its own name, the so-called nonnegativity preservation property If we want to handle the homogeneous Dirichlet boundary condition abstractly, we have to introduce a new bilinear form a0 , formally the same as (2) with the exception that it is defined for u ∈ H01 (Ω), v ∈ H01 (Ω) Then by the discretization we simply not have K∂ Discontinuous Galerkin method  problem setting and discretization Discontinuous Galerkin methods have been thoroughly investigated in recent years [1, 2, 11] These methods have several advantages: • built-in stability for time-dependent advection–convection equations, • adaptivity can be easily done (the basis functions not have to be continuous over the interfaces), • the mesh does not have to be regular, hanging-nodes can be handled easily, • conservation laws could be achieved by numerical solutions For more details see, e.g., [6, 7, 14] The idea behind the discontinuous Galerkin method in comparison with FEM with piecewise linear and continuous basis functions is to get better approximations and to spare computational time by dropping the continuity requirement (even in the case when the solution of the original problem is continuous, as it holds for many applications) ✻✻✼ Discrete maximum principle for interior penalty discontinuous Galerkin methods 4.1 Problem setting Let us set Ω = (0, 1) and consider the following special elliptic operator K , dom K = H (0, 1), defined as K u = −(pu ) + k u, (3) where p, k ∈ R, p > It is clear that for this operator the maximum principle holds due to Theorem 2.2 Here we remark that the space H (0, 1) consists of continuous functions Note that continuity is an important qualitative property and it cannot be preserved by the discontinuous Galerkin method This is one of the reasons why we need to be careful, especially with the preservation of some milder qualitative properties which are in connection with the continuity This leads directly to the investigation of maximum principle for the discontinuous Galerkin method There are several sorts of discontinuous Galerkin methods in the literature In this paper we will consider the interior penalty discontinuous Galerkin method 4.2 The construction of the IPDG elliptic operator As opposed to the standard FEM approach, here the first step to discretize the operator (3) with the interior penalty discontinuous Galerkin method is to define a mesh on (0, 1) Let us denote it by τh and define in the following way: = x0 < x1 < x2 < < xN−1 < xN = We use the notations In = [xn−1 , xn ], hn = |In |, hn−1,n = max {hn−1 , hn } (with h0,1 = h1 , hN,N+1 = hN ) The next step is to define the space Dl (τh ) = {v : v In ∈ Pl (In ), n = 1, 2, , N} – piecewise polynomials over every interval with maximal degree l For these functions we introduce the right and left hand side limits v(xn+ ) = limt→0+ v(xn + t), v(xn− ) = limt→0+ v(xn − t), and jumps and averages over the mesh nodes as [ u(xn )]] = u(xn− ) − u(xn+ ), {{u(xn )}} = (u(xn− ) + u(xn+ )) At the boundary nodes these are defined as {{u(x0 )}} = u(x0+ ), [ u(x0 )]] = −u(x0+ ), [ u(xN )]] = u(xN− ), {{u(xN )}} = u(xN− ) We fix the penalty parameter σ ≥ and ε which can be any arbitrary number, but is usually chosen from the set {−1, 0, 1} The value ε = gives the nonsymmetric, ε = the incomplete, and ε = −1 the symmetric IPDG In [2] several DG methods were examined, and conditions for the convergence were collected The nonsymmetric version converges for all σ > 0, while the two other converge only for σ > σ ∗ , where σ ∗ is unknown for both methods The symmetric method is the only one that guarantees optimal convergence order, because the symmetric version is the only one that is adjoint consistent After these preparations we are ready to define the (discrete) IPDG bilinear form as N−1 xn+1 aDG (u, v) = N pu (x) v (x) dx − n=0 x n N {{pu (xn )}} [ v(xn )]] + ε N + n=0 {{pv (xn )}} [ u(xn )]] n=0 n=0 σ hn,n+1 k uv dx [ v(xn )]] [ u(xn )]] + Note that fixing the parameters σ , ε and the mesh τh can be done in parallel The crucial step is the following We fix a basis in the space Dl (τh ) First we need to choose l = for the same reasons as in the FEM case discussed in Section When choosing the basis functions we need to consider the following If we want to use the Definition 3.1 and apply Theorems 3.2 and 3.3, then we need to choose basis functions with the important properties listed in subsection 3.1 We already set aside continuity, but the next choice fulfils the second and third property and a milder version of the fourth, and this is enough for us ✻✻✽ T.L Horváth, M.E Mincsovics We will use Φ1i (x) for the (2(i − 1) + 1)th basis functions, and Φ2i (x) for the (2(i − 1) + 2)th basis functions, see Figure + On interval Ii the function Φ1i (x) is the linear function with Φ1i (xi−1 ) = 1, Φ1i (xi− ) = and Φ2i (x) is the linear function with + − Φi (xi−1 ) = 0, Φi (xi ) = 1, and these functions are zero outside Ii , see Figure Figure Φ1i (x) and Φ2i (x) Finally, we construct the IPDG elliptic operator similarly to the way we did in the previous section However there are slight differences This matrix can be split in a partitioned form by separating the (discrete) interior and boundary nodes as K0 K∂ K= , A B where K ∈ R(2N)×(2N) , K0 ∈ R(2N−2)×(2N−2) , and the others are trivial The 2N basis functions are ordered as follows: the first 2N − are the basis functions that belong to the interior nodes and they are numbered from left to right The (2N − 1)th belongs to the left boundary and the 2N th belongs to the right boundary Note that the matrices A and B are not important from the point of view of the maximum principle, thus we can omit them So the matrix we need to investigate has the usual form K = [K0 |K∂ ] Remark 4.1 When working with the homogeneous Dirichlet boundary condition we could restrict aDG to D10 × D10 , where D10 (τh ) = {v ∈ D1 (τh ) : v(0) = v(1) = 0} (Φ11 (x) and Φ2N (x) are excluded from the basis), although this is not a usual practice in the discontinuous Galerkin community Let us denote the corresponding bilinear form by a0DG and define it as N−1 a0DG (u, v) xn+1 N−1 pu (x)v (x) dx − = n=0 x n N−1 {{pu (xn )}} [ v(xn )]] + ε n=1 N−1 + n=1 {{pv (xn )}} [ u(xn )]] n=1 σ hn,n+1 k uv dx [ v(xn )]] [ u(xn )]] + In this case the discrete operator simplifies to K0 and, similarly to Remark 3.4, only (T1) or (P1) should be fulfilled In the following we calculate the elements of the matrix K ✻✻✾ Discrete maximum principle for interior penalty discontinuous Galerkin methods 4.3 The exact form of the discrete operators It is easy to check that ∂x Φ1i (x) = −1/hi , ∂x Φ2i (x) = 1/hi , which means that the averages are ∂x Φ1i (xk ) =− , 2hi ∂x Φ2i (xk ) = 2hi at both endpoints xk of Ii , with the exception of the boundary nodes, where there is no division by Similarly, the jumps are Φ1i (xi−1 ) = −1, Φ2i (xi ) = and zero elsewhere Using these facts we can calculate the matrix entries Summing them up we have the following discretization matrices:  d1  t2  w       K0 =           r1 e2 q2 s2 s2 q2 w2 d2 r2 s3 t3 e3 q3 w3 wi qi di si−1 ti  ri ei wN−1          ,         rN−1  si+1 qi wi qN−1 dN−1 sN−1 tN eN  v1  s1  0   K∂ =    0  0  0  0    ,  0  sN  vN where di = ei = wi = qi = ri = si = ti = vi = σ pε hi p + + + k2 , i = 1, , N − 1, 2hi hi,i+1 2hi p σ pε hi i = 2, , N, + + + k2 , 2hi hi−1,i 2hi pε , i = 2, , N − 1, 2hi p p pε hi i = 2, , N − 1, − + − + k2 , hi 2hi 2hi p σ pε − − , i = 1, , N − 1, 2hi+1 hi,i+1 2hi p − , i = 1, , N, 2hi p σ pε − − , i = 2, , N, 2hi−1 hi−1,i 2hi p p pε hi − + − + k , i = 1, , N, hi 2hi hi and zero elsewhere Maximum principle for IPDG elliptic operators Our aim is to get useful mesh conditions that guarantee the discrete maximum principle by using Theorem 3.3 First we deal with (P1) To this aim, we ask for the diagonal elements of the matrix K0 to be nonnegative and the off-diagonal elements to be nonpositive ✻✼✵ T.L Horváth, M.E Mincsovics • di , ei • wi We get the following conditions for ε: ε ≥ −1 − 2σ hi 2k h2i − , phi,i+1 3p i = 1, , N − 1, ε ≥ −1 − 2σ hi 2k h2i − , phi−1,i 3p i = 2, , N wi should be nonpositive, which indicates ε≤0 (4) in the case where we have more than two subintervals See the third part of Remark 5.4 for the degenerate case This means that ε = is excluded generally • qi Because of qi we need to guarantee −p/(2hi ) − pε/(2hi ) + k hi /6 ≤ 0, i = 2, , N − 1, which means the following for ε: ε ≥ −1 + k h2i , 3p i = 2, , N − Or, rephrasing it for the mesh, we have h2i ≤ 3(1 + ε)p/k , i = 2, , N − 1, in the case where k = (In the case k = we simply have ε ≥ −1.) • si The inequality si < always holds • ri , ti We need to guarantee p/(2hi+1 ) − σ /hi,i+1 − pε/(2hi ) ≤ and p/(2hi−1 ) − σ /hi−1,i − pε/(2hi ) ≤ After re-indexing ti and reformulating we have hi,i+1 εhi,i+1 2σ − ≤ hi+1 hi p and hi,i+1 εhi,i+1 2σ , − ≤ hi hi+1 p i = 1, , N − (5) Finally, we show that there is no other restriction needed since the following lemma is valid Lemma 5.1 There exists a positive vector v with K0 v > Proof First let us consider the case where k = and p = We choose the dominant vector v as the piecewise j linear interpolation of the function d(x) = c − x with the bases of Φi in the interior nodes and zero at x = 0, 1, where c ≥ 1, see Figure We prove that this choice is suitable j Let us denote this interpolation by Πd (x) and let v contain its coefficients, so Πd (x) = (i,j)∈int(τh ) v2(i−1)+j−1 Φi (x), where the summation goes over all basis functions with exception of the two that belong to the boundary nodes (Φ11 (x) and Φ2N (x)) It is clear that v > and we need to prove that K0 v > holds The meaning of this inequality is that j aDG (Πd (x), Φi (x)) > holds for all basis functions, since, for example, for the first coordinate of K0 v, (K0 v)1 = j v2(i−1)+j−1 aDG (Φi (x), Φ21 (x)) = aDG (i,j)∈int(τh ) j v2(i−1)+j−1 Φi (x), Φ21 (x) = aDG (Πd (x), Φ21 (x)) (i,j)∈int(τh ) Next we calculate these bilinear forms The function Πd (x) is continuous, therefore its jumps are zero all over the nodes, which means we have to take into account neither ε, nor the penalty terms ✻✼✶ Discrete maximum principle for interior penalty discontinuous Galerkin methods Figure Πd (x) for c = 1.3 The derivative of Πd (x) can be calculated on every In It is c − x12 x1 on I1 , − xi2 − xi−1 = − xi − xi−1 xi − xi−1 on Ii , −c xN−1 − xN−1 i = 2, , N − 1, on IN This means aDG Πd (x), Φ21 (x) = I1 ∂x Πd (x) ∂x Φ21 (x) dx − {{∂x Πd (x1 )}} Φ21 (x1 ) c − x12 x1 = I1 (c − x12 )/x1 − x1 − x2 dx − h1 ·1 = c − x12 x1 + x2 + 2x1 (6) =1 Similarly, aDG Πd (x), Φ12 (x) = c − x12 x1 + x2 + 2x1 For i = 1, N − 1, N, aDG Πd (x), Φ2i (x) = Ii ∂x Πd (x) ∂x Φ2i (x) dx − {{∂x Πd (xi )}} Φ2i (xi ) = −(xi + xi−1 ) xi + xi−1 + xi + xi+1 dx − − I i hi ·1 = xi+1 − xi−1 (7) For i = 1, 2, N, aDG Πd (x), Φ1i (x) = Ii ∂x Πd (x) ∂x Φii (x) dx − {{∂x Πd (xi−1 )}} Φ2i (xi−1 ) xi + xi−1 + xi−1 + xi−2 = −(xi + xi−1 ) − dx − − h i Ii ✻✼✷ xi − xi−2 · (−1) = (8) T.L Horváth, M.E Mincsovics On IN−1 , aDG Πd (x), Φ2N−1 (x) = ∂x Πd (x) ∂x Φ2N−1 (x) dx − {{∂x Πd (xN−1 )}} Φ2N−1 (xN−1 ) IN−1 = − (xN−2 + xN−1 ) =− IN−1 hN−1 dx − x2 − c −(xN−2 + xN−1 ) + N−1 − xN−1 ·1 (9) xN−2 + xN−1 c − xN−1 + 2(1 − xN−1 ) Finally, aDG Πd (x), Φ1N (x) = − c − xN−1 xN−2 + xN−1 + 2(1 − xN−1 ) We have to prove that these are positive values The first three (6)–(8) are trivial To prove that (9) is positive, some simple calculation is still needed − c − xN−1 xN−2 + xN−1 + > 0, 2(1 − xN−1 ) c − xN−1 > xN−2 + xN−1 , − xN−1 and this holds since √ √ c − xN−1 ( c − xN−1 )( c + xN−1 ) √ = > c + xN−1 > + xN−1 > xN−2 + xN−1 − xN−1 − xN−1 When p = 1, we only have to multiply the matrix K0 by p, which makes no difference in the sign of the product When j k = 0, we have the extra terms Ii k Φi (x) · Φli (k), where j, l ∈ {1, 2} All functions are positive, so these integrals are also positive We have just increased the elements of K0 , consequently increased the coordinates of K0 v Accordingly, we can apply Theorem 7.3, which completes the investigation of the condition (P1) Property (P2) means that v1 and vN should be nonpositive, i.e., ε≥ −3p + k h2i k h2i =− + ≥− , 6p 6p i = 1, N (10) Note, it means that ε = −1 is excluded Property (P3) means that ≤ (K0 |K∂ )e should hold It is equivalent to j aDG (1, Φi ) ≥ for (i, j) ∈ int(τh ), for example, for the first coordinate of (K0 |K∂ )e: N N ((K0 |K∂ )e)1 = j · aDG Φi (x), Φ21 (x) = aDG i=1 j=1 j · Φi (x), Φ21 (x) = aDG (1, Φ21 (x)) i=1 j=1 The result of this matrix-vector product is  k h1 p −ε h1 k h2           k hN−1    k 2h p N −ε hN               ✻✼✸ Discrete maximum principle for interior penalty discontinuous Galerkin methods which is nonnegative if ε≤ k h2i , 2p i = 1, N (11) We should note that we need to take it into consideration only in the degenerate case, when the interval is divided into two subintervals, since (4) is stricter Inequalities (10) and (11) can be pulled together as k h2i k h2i ≤ε≤ , − + 6p 2p i = 1, N, (12) 2pε 3p(2ε + 1) ≤ h2i ≤ , k2 k2 i = 1, N (13) or, rephrasing it for the mesh, 5.1 The mesh conditions In this subsection we sum up and systematize the conditions we obtained Our plan is to give a “recipe” on how we should choose the parameters and the mesh to guarantee the discrete maximum principle The trick is that we fix the order of the choices First we suppose that the interval (0, 1) is divided into more than two subintervals Theorem 5.2 Let K = [K0 |K∂ ] be the matrix constructed from (3) by the bilinear form aDG as described in subsection 4.2 This matrix has the discrete maximum principle if we choose • ε so that −1/2 ≤ ε ≤ when k = 0, −1/2 < ε ≤ when k > 0, • σ so that p(1 − ε)/2 ≤ σ , • the mesh τh so that 3p(2ε + 1) , i = 1, N, k2 3p(ε + 1) h2i ≤ , i = 2, , N − 1, k2 εhi,i+1 2σ hi,i+1 εhi,i+1 2σ − ≤ and − ≤ , hi p hi hi+1 p h2i ≤ hi,i+1 hi+1 (fineness at the boundary) (fineness at the interior) i = 1, , N − (uniformity) Proof Almost all the conditions are simple consequences of the above calculations The condition for σ can be derived from (5) by taking its minimum 2σ hi,i+1 εhi,i+1 ≥ − ≥ − ε p hi+1 hi Note that we have two types of mesh conditions, one is about the fineness of the mesh and the other is about the uniformity The first determines the maximum size of the subintervals and it depends on the choice of ε, ε = is the least restrictive The second determines the maximum ratio of the size of the neighboring subintervals and it depends on the choice of σ , σ = p(1 − ε)/2 is the most restrictive When working with homogeneous Dirichlet boundary conditions, we have only to fulfil (P1), see Remark 4.1 This leads to the following conditions ✻✼✹ T.L Horváth, M.E Mincsovics Theorem 5.3 Let K = K0 be the matrix constructed from (3) by the bilinear form a0DG as described in subsection 4.2 This matrix possesses the discrete maximum principle if we choose • ε so that −1 ≤ ε ≤ when k = 0, −1 < ε ≤ when k > 0, • σ so that p(1 − ε)/2 ≤ σ , • and the mesh τh with 3p(ε + 1) , k2 2σ ≤ and p h2i ≤ εhi,i+1 hi,i+1 − hi+1 hi i = 2, , N − 1, (fineness at the interior) hi,i+1 εhi,i+1 2σ − ≤ , hi hi+1 p i = 1, , N − (uniformity) Remark 5.4 In addition, we investigate the most popular cases: ε ∈ {−1, 0, 1} ε = −1 We can guarantee the discrete maximum principle in this case only if k = holds and a0DG is used as a discretization In this case (5) simplifies to the following: hi,i+1 hi,i+1 2σ , + ≤ hi hi+1 p i = 1, , N − (14) This has the consequence that σ needs to be chosen ≥ p ε = We have no additional restrictions in this case The conditions simplify as hi,i+1 2σ ≤ hi+1 p and hi,i+1 2σ , ≤ hi p i = 1, , N − 1, which can be pulled together as hi,i+1 2σ ≤ , {hi , hi+1 } p i = 1, , N − 1, (15) since it is enough to guarantee that the inequality holds for the greater left-hand side Thus σ has to be chosen ≥ p/2 ε = We can guarantee the discrete maximum principle in this case only if (0, 1) is subdivided into two subintervals Then (5) leads to the following conditions: h1,2 h1,2 2σ − ≤ h1 h2 p and h1,2 h1,2 2σ − ≤ h2 h1 p They can be pulled together as h1,2 − {h1 , h2 } 2σ ≤ {h1 , h2 } p (16) Then discretization aDG is used, we have more conditions, namely k > and 2p 9p ≤ h2i ≤ , k2 k i = 1, Remark 5.5 If we choose a different definition for hn−1,n , namely, hn−1,n = {hn−1 , hn }, c.f [6, Chapter 4, Definition 4.5] and [14, Chapter 1], the condition for σ will coincide with the condition that describes the relation between the neighboring subintervals ✻✼✺ Discrete maximum principle for interior penalty discontinuous Galerkin methods Numerical examples and conclusion 6.1 Numerical examples  on the sharpness of the conditions In this section we will investigate the mesh conditions we derived Naturally, these cannot be sharp since we applied Theorem 3.3, whose conditions are only sufficient and not necessary However, we will show that we obtain sharpness in some sense Example 6.1 Let us set p = 1, ε = 0, σ = 5, k = First, it is clear that condition (12) holds for ε and (13) is out of view In this case for the mesh τh = {0, 0.02, 0.22, 0.8, 1} the condition (15) is sharp in the following sense Let us modify this mesh as τhm = 0, 0.02, 0.22 + , 0.8, 10m Let us consider the vector v = [−1, 1/10m , 0, 0, 0, 0]T , see Figure The following calculation shows that the resulting right-hand side is nonpositive, which means that the maximum principle fails Figure Left: the counterexample with m = Right: the positive value at the node 0.221 The product Kv has only four nonzero coordinates: [−d1 + r1 /10m , −t2 + e2 /10m , −w2 + q2 /10m , s2 /10m , 0, 0, 0, 0]T In this case h1,2 = h2 Let us examine these terms −d1 + r1 = − − + m 10m 2h1 h2 10 − 2h2 h2 = − − − m < 2h1 h2 10 2h2 The second one is −t2 + e2 = − + + m 10m 2h1 h2 10 + 2h2 2h2 = −25 + h2 5+ 11 · 10m Simple computations prove its negativity: 5(10n+1 + 11) + 11/(2 · 10m ) = = 2(10n + 5) 1/5 + 1/10m h2 5(10n+1 + 11) < 50(10n + 5), ✻✼✻ 5+ 11 · 10m < 25, · 10n+1 + 55 < · 10n+1 + 250 T.L Horváth, M.E Mincsovics The last two terms are easier: −w2 + q2 q2 =0+ m = m 10m 10 10 − 1 + h1 2h1 =− < 0, · 10m · h1 s2 1 = m − 10m 10 2h1 < Example 6.2 Let us set p = 1, ε = 1, σ = 5, k = and use a0DG In the case that was discussed in the third part of Remark 5.4 the mesh τh = {0, 1/12, 1} is sharp in the same sense as in the last example with respect to (16) Similarly as above, we modify the mesh as τhm = {0, 1/12 − 1/10m , 1} and choose v = [−1, 1/10m ]T which breaks the maximum principle Then K0 v = [−d1 + r1 /10m , −t2 + e2 /10m ]T , where 1 r1 = − − − + m 10m h1 h2 2h1 10 e2 1 −t2 + m = − + + + m 10 h1 h2 2h2 10 −d1 + 5 1 − − = − − − + m − − 2h2 h2 h1 h1 h2 2h1 10 2h2 h1 5 1 + + = − + + + m 2h2 h2 2h2 h1 h2 2h2 10 h2 < 0, Similar calculations as before give 1 − m 12 10 1 + + m < , h2 2h2 10 h2 h1 12 11 11 + m < + m, 10 12 10 12 < h2 , 10m 11 11 12 11 + m − m − 2m < + m, 12 10 10 10 12 10 h1 11 + which holds for all m > 6.2 Conclusion First, we have shown that it is possible to guarantee the discrete maximum principle when IPDG discretization is used However, we should mention that our conditions are restrictive at the following points: • the choice of the basis functions, • ε = is excluded from a practical point of view, • we can handle ε = −1 only in special cases On the other hand, we could state that ε = works very well from the discrete maximum principle point of view and the conditions suggest that we need to take into consideration a non-integer ε ∈ (−1/2, 0) We have shown with numerical examples that our conditions are sharp in some sense The numerical examples and computational tests suggest the following points of interest: • for the symmetric IPDG, (14) does not seem to be sharp, • the mesh condition (15) seems to be sharp only at the boundary, it could be slightly broken in the interior intervals without losing the maximum principle, • for meshes that consist of more than two subintervals, the condition (16) seems to be irrelevant for the neighboring elements ✻✼✼ Discrete maximum principle for interior penalty discontinuous Galerkin methods Appendix: M-matrices The M-matrix theory provides a powerful tool to prove that a matrix is inverse nonnegative This subsection is based on [3, Chapter 6] with small changes Definition 7.1 We call a real matrix Z-matrix if its off-diagonal entries are nonpositive Definition 7.2 We call a real matrix M-matrix if it can be represented as sI − B, where I is the identity matrix and B ≤ 0, moreover s ≥ ρ(B), where ρ denotes the spectral radius of a matrix It is obvious that an M-matrix is also a Z-matrix Theorem 7.3 ([3, Chapter 6, Theorem 2.3]) We assume that the matrix A is a Z-matrix Then the following statements are equivalent A is a nonsingular M-matrix There exists u > with Au > There exists A−1 and A−1 ≥ References [1] Ainsworth M., Rankin R., Technical Note: A note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes, Numer Methods Partial Differential Equations, 2012, 28(3), 1099–1104 [2] Arnold D.N., Brezzi F., Cockburn B., Marini L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal., 2001/02, 39(5), 1749–1779 [3] Berman A., Plemmons R.J., Nonnegative Matrices in the Mathematical Sciences, Comput Sci Appl Math., Academic Press, New York–London, 1979 [4] Ciarlet P.G., Discrete maximum principle for 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approximate problems of nonpositive type, J Fac Sci Univ Tokyo Sect IA Math., 1982, 29(2), 473–491 [16] Vejchodský T., Discrete Maximum Principles, habilitation thesis, Institute of Mathematics of the Academy of Sciences and Faculty of Mathematics and Physics, Charles University, Prague, 2011 [17] Vejchodský T., Šolín P., Discrete maximum principle for higher-order finite elements in 1D, Math Comp., 2007, 76(260), 1833–1846 ✻✼✾ ... 3.4 ✻✻✺ Discrete maximum principle for interior penalty discontinuous Galerkin methods The following step is to define a mesh on Ω A 1D mesh consists of intervals The discrete maximum principle. .. of maximum principle for the discontinuous Galerkin method There are several sorts of discontinuous Galerkin methods in the literature In this paper we will consider the interior penalty discontinuous. .. the original problem is continuous, as it holds for many applications) ✻✻✼ Discrete maximum principle for interior penalty discontinuous Galerkin methods 4.1 Problem setting Let us set Ω = (0,