DISCRETE MAXIMUM PRINCIPLES in THE FINITE ELEMENT SIMULATIONS

Transcription

1 DISCRETE MAXIMUM PRINCIPLES in THE FINITE ELEMENT SIMULATIONS Sergey Korotov BCAM Basque Center for Applied Mathematics 1

2 The presentation is based on my collaboration with several colleagues - I will mention their names (and papers) in the appropriate places of the talk. 2

3 Model Elliptic Problem Find a function u such that div(a u)+cu = f in Ω, (1) u = g on Ω. (2) Ω R d is a bounded polytope with Lipschitz boundary Ω. The diffusive tensor A is assumed to be a symmetric and uniformly positive definite matrix. The reactive coefficient c is assumed to be nonnegative in Ω. 3

4 Continuous Maximum Principle The classical (smooth) solutions of many elliptic/parabolic problems are known to satisfy various (continuous) maximum principles (or CMPs in short). For our model elliptic problem standard CMP reads as follows: f 0 = max x Ω u(x) max{0, maxg(s)}. (3) s Ω CMPs are not only pure mathematical properties of PDE models, they also reflect well physical behaviour of phenomena modelled. 4

5 Preliminaries Some reasonable discrete analogue of CMP (which may depend, in general, on the nature of numerical technique used) is often called the discrete maximum principle (or DMP in short). Conditions on parameters of computational schemes (e.g. on mesh shape and size, time-step values, etc) a priori providing a validity of DMPs by computed approximations are highly desired In what follows we shall only consider elliptic problems 5

6 The first DMP and certain conditions providing its validity were (probably?) formulated by R. Varga for the finite difference method (FDM) in On a discrete maximum principle, J. SIAM Numer. Anal. 3 (1966), However, only a special case f = 0 was considered there and CMP was of the following form: max x Ω u(x) max s Ω g(s). 6

7 Later, in 1970 Ph. Ciarlet presented a more sophisticated form of DMP suitable for both, finite element (FE) and finite difference (FD) types of discretization. He also proposed a practically convenient set of (sufficient) conditions on matrix blocks involved, which provides a validity of his DMP. Since that time Ciarlet conditions became popular in numerical community for proving DMPs for various problems of elliptic type. We shall present the conditions proposed by Ciarlet, sketch appearing practical problems with them when FEM is used as a main numerical technique and discuss some generalizations. 7

8 Weak Formulation Standard (linear) FE schemes are based on weak formulations. For our test problem it reads as follows: Find u g +H 1 0(Ω) such that a(u,v) = F(v) v H 1 0(Ω), where a(u,v) = Ω A u vdx+ Ω cuvdx and F(v) = Ω fvdx. Here, the matrix A is assumed to be in [L (Ω)] d d, c L (Ω), g H 1 (Ω), and f L 2 (Ω). The existence and uniqueness of the weak solution u is provided by the standard Lax-Milgram lemma. 8

9 FE Discretization: Mesh & Basis Functions Let T h be a FE mesh of Ω with interior nodes B 1,...,B N lying in Ω and boundary nodes B N+1,...,B N+N lying on Ω. Let V h be a finite-dimensional subspace of H 1 (Ω), associated with T h and its nodes, being spanned by basis functions φ 1,φ 2,...,φ N+N s.t. φ i 0 in Ω, i = 1,...,N +N, and N+N i=1 φ i 1 in Ω. We also assume that φ 1,φ 2,...,φ N vanish on the boundary Ω. Thus, they span a finite-dimensional subspace V 0 h of H 1 0(Ω). 9

10 FE Discretization: Matrix Equation Let g h = N+N i=n+1 g iφ i V h be a suitable approximation of the function g, for example its nodal interpolant. FE approximation is defined as a function u h g h +V 0 h such that a(u h,v h ) = F(v h ) v h V 0 h. In fact, u h = N+N i=1 y i φ i, where ȳ = [y 1,...,y N+N ] is the solution of of the following square system of equations Āȳ = F, (4) with Ā = A A 0 I, F = F F. (5) 10

11 Ā = A A 0 I, F = F, and Ā 1 = F A 1 A 1 A 0 I. Blocks A and A are matrices of size N N and N N, respectively, I stands for the unit matrix, and 0 for the zero matrix. Entries of Ā are a ij = a(φ j,φ i ), i = 1,...,N, j = 1,...,N +N. The block-vector F consists of entries f i = F(φ i ), i = 1,...,N, and the block-vector F has entries f i = g i, i = N +1,...,N +N. For the later reference we include the formula for Ā 1. Ā is nonsingular if and only if A is nonsingular. 11

12 Sufficient Algebraic Conditions of Ph. Ciarlet We should distinguish two basically different types of DMPs. Algebraic DMP: A natural algebraic analogue of CMP is F 0 = max y i max{0, max y j }. i=1,...,n+n j=n+1,...,n+n Functional DMP: A natural functional imitation of CMP is f 0 = max Ω u h max{0,max Ω u h}. It is easy to see that the above types of DMPs are actually equivalent e.g. for linear, multilinear, or prismatic FEs. (However, these DMPs are not equivalent, in general, for higher-order FEs.) 12

13 Ā = A A 0 I, F = F F, and Ā 1 = A 1 A 1 A 0 I. In the paper Discrete maximum principle for finite-difference operators, Aequationes Math. 4 (1970), , Ciarlet proved Theorem 1. The algebraic DMP is satisfied if and only if (A) Ā is monotone (i.e., Ā nonsingular and Ā 1 0) (B) ξ +A 1 A ξ 0, where ξ and ξ are vectors of all ones of sizes N and N, respectively. Since conditions (A) and (B) are difficult to verify in practice, Ciarlet proposed in the same work a more convenient set of sufficient conditions. 13

14 Ā = A A 0 I, F = F F, and Ā 1 = A 1 A 1 A 0 I. Theorem 2. The algebraic DMP is valid provided the matrix Ā satisfies (a) a ii > 0, i = 1,...,N, (b) a ij 0, i j, i = 1,...,N, j = 1,...,N +N, a ij 0, i = 1,...,N, (c) N+N j=1 (d) A is irreducibly diagonally dominant. Ciarlet essentially proposed the above conditions in order to utilize the following result of R. Varga from his book Matrix Iterative Analysis, 1962: 14

15 Lemma 3. If A R N N is an irreducibly diagonally dominant matrix with strictly positive diagonal and nonpositive off-diagonal entries then A 1 > 0. Ā = A A 0 I, F = F, and Ā 1 = F A 1 A 1 A 0 I. Remark 1. However, in the proof we only need monotonicity of A, i.e. A 1 0, not more. Remark 2. In case g = 0, the system reduces to Ay = F, and CMP is f 0 = max x Ω To immitate this, we also need A 1 0, not more. u(x) 0. (6) 15

16 Analysis of Ciarlet Conditions Condition (a) is always satisfied for elliptic problems. The row sums in (c) are nonnegative automatically for our test problem, because the basis functions are nonnegative and form a partition of unity: N+N j=1 a ij = a ( N+N j=1 φ j,φ i ) = a(1,φ i ) = Ω cφ i 0, i = 1,...,N. (7) On the other hand, the irreducibility of A required in (d) is not always obvious. 16

17 Problems with Irreducibility Figure 1: [Hannukainen, Korotov, Vejchodský, 2009]. Meshes leading to reducible matrix A for the Poisson problem with Dirichlet boundary conditions. Angles with dots are right. 17

18 Associated Geometric Conditions For some types of FEs, Ā can be computed explicitly, therefore condition (b): a ij 0, i j, can often be guaranteed a priori by imposing suitable geometrical requirements on the shape and size of FE meshes. For example, let A = I, then there are some known geometrical conditions providing (b). 18

19 Simplicial FE Meshes For simplicity we assume that c is constant and d = 2 If nodes B i and B j are not connected by an edge then a ij = 0 B i T 1 α 2 T 2 α 1 B j Otherwise, a ij always consists of two contributions computed over two (adjacent) triangles from suppφ i suppφ j : a ij = 1 ) (ctgα 1 +ctgα 2 + c ) (meast 1 +meast [Fujii, 1973], [Ruas Santos, 1982], [Ciarlet-book, 1978]. 19

20 a ij = 1 2 (ctgα 1 +ctgα 2 ) + c ) (meast 1 +meast 2 12 In case c = 0, we get a ij 0 if ctgα 1 +ctgα 2 0 It holds if e.g. α k π 2. This observation leads to the concept of nonobtuse triangulations In case c > 0, we get a ij 0 if c ) (meast 1 +meast 2 1 ) (ctgα 1 +ctgα It is possible if e.g. α k π 2 ε (ε > 0) and triangulations are sufficiently fine. This leads to the concept of acute triangulations 20

21 Similarly for d 3: B i B j K r B i S ir B j S jr α K r ij φ j φ i dx = meas d 1S ir meas d 1S jr K r d 2 meas d K r d! φ j φ i dx = K r (d+2)! meas dk r cosα K r ij [Křížek, Lin Qun, 1995], [Xu, Zikatanov, 1999], [Brandts, Korotov, Křížek, 2007] 21

22 Block FE Meshes [Karátson, Korotov, Křížek, 2007], [Korotov, Vejchodský, 2010]: the results strongly depend on dimension Let K be a block of a d-dimensional block mesh with edges of lengths b 1,b 2,...,b d, and let B i and B j be its vertices connected by b 1, then a K ij = b 1b 2...b d 3 d 1 ( d k=2 1 2b 2 k 1 b 2 1 ), i j d = 2: all rectangular elements have to be nonnarrow for (b), i.e. 2/2 b1 /b 2 2 d = 3: trilinear elements have to be cubes d 4: condition (b) is not valid even on hyper-cubes 22

23 Prismatic FE Meshes [Hannukainen, Korotov, Vejchodský, 2009]: for 3D meshes consiting of right triangular prisms the altitudes of prisms should be limited from both sides by certain quantities dependent on the area and angles of the triangular base (and the magnitude of coefficient c) to provide (b). F D E C d b γ a α β A c B 23

24 No Irreducibility-Proof is Needed A real square matrix A is an M-matrix if all its off-diagonal entries are nonpositive and if it is nonsingular and A 1 0. A real square matrix A is a Stieltjes matrix if all its off-diagonal entries are nonpositive and if it is symmetric and positive definite. The following lemma from [Varga-book, 1962] enables to eliminate (a), (c), and (d) from the set of Ciarlet conditions in our case: Lemma 4. If A is a Stieltjes matrix then it is also an M-matrix. Theorem 5. If the finite element matrix Ā, associated to our model problem, satisfies condition (b), a ij 0, i j, then the algebraic DMP is valid. For the proof see [Hannukainen, Korotov, Vejchodský, 2009]. 24

25 Geometrical Conditions are Essential [Brandts, Korotov, Křížek, Šolc, 2009]: one single obtuse triangle in the FE mesh can destroy DMP (e.g. for the Poisson equation with zero Dirichlet b.c.) 25

26 Problems with Mesh Reconstructions To improve the accuracy of FE computations we need to refine meshes globally and locally. How to preserve the desired geometric properties then? red refinement For global mesh refinements 2d red technique always preserves properties of acuteness and nonobtuseness 26

27 For local mesh refinements green post-refining (to preserve conformity) is often needed, but it generally leads to obtuse angles. Only in special cases we can preserve nonobtuseness for local mesh refinements using both, red and green, refinements: red refinement green refinement red + green producing nonobtuse local refinements 27

28 Problems with 3D FE meshes Existence of a face-to-face partition of a cube into acute tetrahedra was surprisingly (!) an open problem till D red (global) refinement technique does not generally preserve nonobtuseness or acuteness. green refinement red green refinement Problems with local refinements (especially for acuteness)... 28

29 More Problems Local refinements of block and prismatic FE meshes... Full diffusive tensor... Higher-dimensional problems (blocks?, acute simplices?)... 29

30 THANK YOU FOR YOUR ATTENTION! 30

31 On Sharpness of Theoretical Conditions Stieltjes matrices form only a certain subclass of monotone matrices. In what follows we test how sharp the conditions based on the Stieltjes matrix concept are, i.e., we try to test how wide is the class of meshes which lead to A being monotone but not Stieltjes. For this purpose we solve the 2D Poisson problem with homogenous Dirichlet boundary conditions on various domains using various triangulations. We present three representative tests... 31

32 In each test we construct a simple triangulation which is characterized by two parameters (angles) 0 < α < π and 0 < γ < π: γ α α α+γ α α α α γ γ γ γ γ Figure 2: The basic meshes for the three tests. The meshes used for the actual computations are 10-fold refined basic meshes. 32

33 Figure 3: Domain 1: triangulations with non-obtuse maximal angle (i.e. matrix A is a Stieltjes matrix). Domain 2: triangulations with obtuse maximal angle providing the DMP (i.e. matrix A is monotone but not Stieltjes). Domain 3: triangulations with obtuse maximal angle, DMP is not valid (matrix A is not monotone). 33

34 Conclusions If we compare the sizes of domains 1 with domains 2 we may conclude that the standard sufficient condition (b) is not very sharp. There is a wide space for its generalization. However, any generalization have to utilize more general criteria for the monotonicity of a matrix. These criteria are often quite complicated and their application for proving DMPs is not so straightforward. 34