Linear Finite Element Methods

Transcription

1 Chapter 3 Linear Finite Element Methods The finite element methods provide spaces V n of functions that are piecewise smooth and simple, and locally supported basis function of these spaces to achieve good approximations of the solution u V, an efficient assembly of the system matrix with desirable properties (e g sparse and/or well conditioned) 31 Meshes A mesh M of the computational domain Ω R d, d = 1,, 3, is a collection of non-overlapping cells { i } M i=1, M := M, that are open, simply connected subsets i Ω Each cell i is a C -diffeomorphic image of a closed d-dimensional polytope the reference cell which is a convex hull of at least d + 1 points in R d Each cell i is an interval (d = 1), a Lipschitz polygon (d = ), or a Lipschitz polyhedron (d = 3) The mesh M contains a set of nodes (vertices) N (M), edges E(M) (d > 1), and faces F(M) (d = 3) (Sometimes the components of the mesh of dimension d 1 are called faces) The topology of the mesh is described by the connectivity of cells, faces, edges and nodes, e g, does a face belong to a cell, or more precisely, is a face contained in the closure of a cell A mesh is conforming if the intersection i j of (the closure of) any two adjacent cells i, j is a face of both cells A node of a mesh M that is located in the interior of a geometric face of one of its cells is known as hanging (dangling) node Grids are meshes with translation invariant structure These can be tensor product grids, that are meshes whose cells are quadrilaterals (d = ) or hexahedra (d = 3) with parallel sides Meshes are a crucial building block in the design of the finite dimensional trial and test spaces used in the finite element method, and provide subdomains for integration to build the system matrix and vector of the right hand side The orientation of geometric objects of the mesh is important for defining high order basis functions For an edge we have to define its direction For a face (d = 3) we have to specify an ordering of the edges along its boundary 5

2 (a) (b) (c) Figure 31: (a) Two-dimensional mesh and the sets of edges (red) and nodes (blue) (b) A geometrical node is not a topological node (c) A triangulation that is not conforming and possesses two hanging nodes Figure 3: Examples of triangular and quadrilateral grids in two dimensions Reference cell for triangles: reference triangle with nodes (, ), (1, ), (1, 1) for quadrilaterals: reference quad [, 1] or [, 1] for tetrahedra: reference tetrahedron with nodes (,, ), (1,, ), (, 1, ), (, 1, ) for prism: reference prism = reference triangle [, 1] for pyramid: reference pyramid with nodes (,, ), (1,, ), (1, 1, ), (, 1, ) and (5, 5, 1) for hexahedra: reference hexahedron [, 1] 3 In meshes with different cell types, we have more than one reference cell (a) Triangle (b) Quadrilateral (c) Tetrahedron (d) Prism (e) Pyramid (f) Hexahedron Figure 33: Cell types in D and 3D Affine mapping: linear, bijective mapping from a reference cell (triangle/parallelogram) to a physical cell (triangle/parallelogram) in the mesh Φ : R d R d, ξ Fξ + τ, F R d,d regular, τ R d (31) A mesh is called affine equivalent if all its cells arise as affine images of a single d-dimensional (reference) cell 6

3 Automatic mesh generation: can be given algorithms that create a mesh starting from a description of Ω which in terms of geometric primitives (ball, brick, etc) whose unions or intersections constitute Ω, by means of a parametrization of the faces of Ω, through a function f : R d R, whose sign indicates whether a point is located inside Ω or outside, by a mesh covering the surface of Ω and a direction of the exterior unit normal Various strategies can be employed for automatic grid generation: advancing front methods, that build cells starting from the boundary, Delaunay refinement techniques, that can create a mesh starting from a mesh for Ω or a cloud of points covering Ω, the quadtree (d = ) or octree (d = 3) approach, which fills Ω with squares/cubes of different sizes supplemented by special measures for resolving the boundary, and mapping techniques, that split Ω into subdomains of simple shape (curved triangles, parallelograms, bricks), endow those with parametric grids and glue these together Remark 31 Traditional codes for the solution of boundary value problems based on the finite element method usually read the geometry from a file describing the topology and geometry of the underlying mesh Then an approximate solution is computed and written to a file in order to be read by post-processing tools like visualization software, see Figure 34 Parameters Mesh generator Finite element solver (computational kernel) Post-processor (e g visualization) Figure 34: Flow of data in traditional finite element simulations Remark 3 A typical file format for a mesh of a simplicial conforming triangulation of a twodimensional polygonal domain is the following: # Two-dimensional simplicial mesh N N ξ 1 η 1 ξ η ξ N η N M N # Number of nodes # Coordinates of first node # Coordinates of second node # Coordinates of N-th node # Number of triangles n 1 1 n 1 n 1 3 X 1 # Indices of nodes of first triangle n 1 n n 3 X # Indices of nodes of second triangle n M 1 n M n M 3 X M # Indices of nodes of M-th triangle Here, X i, i = 1,, M, is an additional piece of information that may, for instance, describe what kind of material properties prevail in triangle #i In this case X i may be an integer index into a look-up table of material properties or the actual value of a coefficient function inside the triangle 7

4 Additional information about edges located on Ω may be provided in the following form: N # Number of edges on Ω n 1 1 n 1 Y 1 # Indices of endpoints of first edge n 1 n Y # Indices of endpoints of second edge n 1 n Y # Indices of endpoints of -th edge where Y k, k = 1,,, provides extra information about the type of boundary condition to be imposed on edge k Note that the ordering of the nodes in the above file formats implies an orientation of triangles and edges For a comprehensive account on mesh generation see [FG] An interesting algorithm for Delaunay meshing is described in [Rup95, She96] Free mesh generation software is also available, just to name some, netgen, gmsh, triangle, emc However, the most sophisticated mesh generation tools are commercial products and their algorithmic details are classified 3 Linear finite elements on triangular meshes 31 Basis functions 1 Figure 35: Finite element basis function b P (x) Let the domain Ω R be a polygon with a triangular conforming mesh M We define the finite element space of piecewise linear, continuous functions as S 1 (Ω, M) := {u C (Ω) : u(x) = a + bx 1 + cx, Proposition 33 (Properties of S 1 (Ω, M)) It holds S 1 (Ω, M) H 1 (Ω) P M} u S 1 (Ω, M) is uniquely defined by the values of u(p ) on the nodes P N (M) N = dim S 1 (Ω, M) = N < S 1 (Ω, M) = span{b P (x) : P N (M)}, with so called hat functions defined as b P S 1 (Ω, M), b P (P ) = δ P =P Let b be the vector of the basis functions Then, an arbitrary FE function v S 1 (Ω, M) can be written as v(x) = v(p )b P (x) = v b(x), so also the solution u n (x) = P N (M) P N (M) u n (P )b P (x) = u b(x) 8

5 3 Assembling of system matrix and load vector The support of the basis functions b P (x) consists of only a few triangles, such that the integrals in the (bi-)linear forms reduce to smaller sets and vanish for many pairs of basis functions Changing the point of view, we consider the shape functions, which are the restrictions of basis functions to one cell M For S 1 (Ω, M) these are exactly three, each for one node of The shape functions can be defined on a single cell as N,Pj()(x) = N j (Φ x), where N j are the element shape functions defined by N (ξ) = 1 ξ 1 ξ, Nj (ξ) = ξ j, j = 1,, on the reference element, which is the triangle with vertices (, ), (1, ), (, 1) P j () is the j-th node of the triangle, j =, 1,, with the coordinate p j Then, the (affine) element mapping reads x = Φ (ξ) = p + ξ 1 (p 1 p ) + ξ (p p ) = τ + F ξ, with F = (p 1 p, p p ) and τ = p As the element mapping is linear, the shape functions are in fact linear as well The basis functions result from the shape functions by glueing, i e, { N,Pj (x) x, P N (), P = P j (), b P (x) = otherwise This can be expressed by the connectivity matrices or T-matrices { T R 3,N 1, P j () = P i (M),, (T ) ij =, otherwise These T-matrices give a relation between the local number of (element) shape functions and global numbering of basis functions They have the form 1 T = 1 1 For the T-matrices an extremely sparse format is used For linear finite elements on triangles we have only to store the three indices P i, P i1 and P i of the three nodes in the triangle Now, we can express a basis function in one cell as and the system matrix is given with b Pi (x) = 3 (T) ki N,Pk ()(x), k=1 (B) ij = b(b Pj, b Pi ) = b (b Pj, b Pi ) = = = 3 b ( (T ) kj N,Pk (), 3 k=1 k=1 l=1 3 (T ) li N,Pl ()) l=1 3 (T ) kj (T ) li b (N,Pk (), N,Pl ()) (T ),jb (T ),i, i e, as sum of weighted element matrices B = T B T This means we have to integrate only over the shape functions, and to sum up the contributions over all the cells 9

6 Element stiffness matrix The bilinear form in the variational problem related to (14) consists of two parts, which contribute to the system matrix The first part a(u, v) = a u, v dx is constituting the stiffness matrix A We transform the derivatives to By chain rule of differentiation and with Ω we have and thus = ( x 1, x ), = ( ξ 1, ξ ), ξ1 N,Pj (x(ξ)) = x1 N,Pj (x) x 1 ξ 1 + x N,Pj (x) x ξ 1 ξ N,Pj (x(ξ)) = x1 N,Pj (x) x 1 ξ + x N,Pj (x) x ξ ) ( Φ N,Pj (x(ξ)) = N,Pj (x) ξ }{{} DΦ =F Hence, for triangular cells we obtain N,Pj (x) = F N,Pj (x(ξ)) = F N j (ξ) For a single cell the entries of the element stiffness matrix A are given by a (N,Pj, N,Pi ) = a(x)( N,Pj (x)) N,Pi (x)dx = a(φ ξ)( N j (ξ)) F F N i (ξ) F dξ = a(φ ξ)( N j (ξ)) adj(f )adj(f ) Ni (ξ) F dξ, where we used the relation of the inverse and the adjoint matrix F = F adj(f ) ( ) ( ) a b d b The adjoint matrix of a matrix is c d c a Note, that the gradient of the element shape functions are constant vectors ( ) ( ) ( ) N 1 =, N1 =, N = 1 So we can simplify (A ) ij = a (N,Pj, N,Pi ) = a ( N j ) adj(f )adj(f ) Ni F with the average heat conduction a = 1 a(x)dx in (Note that F = ) The latter simplifies for constant material or may be obtained by numerical quadrature for more general smooth functions Due to the special values of N j we can even simplify A = a 4 D D with a matrix D with coordinate differences ( ) y1 y D = y y y y 1 x x 1 x x x 1 x 3

7 Element mass matrix The mass matrix is related to the bilinear form m(u, v) = Ω c uv dx The element mass matrix M of the cell can be computed as m (N,Pj, N,Pi ) = c(x) N,Pj (x)n,pi (x) dx = In case of a constant function c in the cell we can write { 1 (M ) ij = 6 c i = j, 1 1 c i j Element load vector The element load vector is related to the linear form l (v) = fvdx c(φ ξ) N j (ξ) N i (ξ) F dξ For general, smooth function f we use numerical quadrature to evaluate the integrals quadrature rule is fvdx f(x )v(x ) The simplest where x is the barycenter of the cell As this quadrature rule is only exact for linear functions, and the shape functions are already linear, it will provide only reasonable results if f is (almost) constant in 33 Numerical quadrature Numerical quadrature for triangular cells are defined on the triangle with nodes (, ), (1, ), (, 1), where integrals over can easily transformed to 1 1 ξ1 f(ξ) dξ dξ 1 = ξ1 f( ξ1+1, ξ+1 ) dξ dξ n j=1 w j f( ξ1,j+1, ξ,j+1 ) There are Gauß quadrature rules for triangles which are exact for polynomials of maximal total degree 1,,, 5 and which use only 1, 3, 4, 6 and 7 points, respectively See, e g, page 141 in Šolín [Šol6] To get higher accuracies in a systematic matter there is, for example, the Duffy transformation, which transforms an integral over the reference triangle by an integral over a reference square and employs a tensor-product quadrature rule After applying the Duffy transformation, we integrate over ξ ξ ξ 1 = ξ 1 (1 ξ ) ξ = ξ ξ 1 ξ 1 Figure 36: Duffy transformation of a square into a triangle 31

8 the reference cell [, 1], so a tensor product of 1D quadrature rule can be applied The quadrature rules in 1D are, however, usually given in the interval [, 1] Hence, we have to transform 1D integrals via 1 f(ξ) dξ = 1 1 f( ξ+1 ) dξ 1 n j=1 w j f( ξj+1 ), where w j are the weights and ξ j the abscissas of the quadrature rule Accurate quadrature rule for smooth functions are variants of the Gauß quadrature The most wellknown is the Gauß-Legendre rule for which the abscissas are the zeros of the n-th Legendre polynomial P n (ξ) and the weights are given by (see [AS64, page 887]) w j = (1 ξ j )[P n(ξ j )] The Gauß-Legendre rule is exact for polynomials of degree n 1 and the remainder (for the interval [, 1]) is R n = (n!) 4 (n + 1)[(n)!] 3 f (n) (ξ), < ξ < 1 The zeros of the Legendre polynomials are tabulated (see [AS64, page 91ff]) and there is an algorithm (see Matlab version gaulegm on the webpage of the lecture) in the numerical recipes [PTFV7] The Gauß-Lobatto rule is only accurate for polynomials of degree n 3 However, both end-points are included in the set of abscissas For the square [, 1] we have the product quadrature rule 1 1 f(ξ) dξ 1 dξ = f( ξ1+1, ξ+1 ) dξ 1 dξ 1 4 n 1 n w i i=1 j=1 w j f( ξi+1, ξj+1 ), with n 1 and n quadrature points in the two directions It is important to note that the Duffy transformation changes the total polynomial degree To see this point, let us start with a definition Definition 34 Given a domain R d, d N, we write P m () := span{ξ ξ α := ξ α1 1 ξ α d d, α Nd, α m} for the vector space of d-variate polynomials of (total) degree m, m N If m = (m 1,, m d ) T N d we designate by Q m () := span{ξ ξ α1 1 ξ α d d, α k m k, 1 k d} the space of tensor product polynomials of maximal degree m k in the k-th coordinate direction Let = convex{ ( ), ( 1 ), ( 1 1) } and =], 1[ Then f(ξ 1, ξ ) dξ = f( ξ 1 (1 ξ ), ξ ) (1 ξ ) d ξ If f P m ( ), then the integrand on the right hand side will belong to Q m,m+1 ( ), i e the degree of polynomial is increased by one in the second variable The usage of numerical quadrature inevitably introduces another approximation, which will contribute to the overall discretization error The general rule is that The error due to numerical quadrature must not dominate the total discretization error in the relevant norms Remark 35 An alternative to numerical quadrature is polynomial interpolation followed by analytical evaluation of the localized integrals 3

9 34 A-priori estimate of the discretization error If the variational problem is elliptic (see Céa s lemma 8) the discretization error is bounded quasi optimally by the best approximation error u u n V b γ n inf u w n w V (3) n W n In case of more general bilinear forms that satisfy (IS1), (IS) and (DIS) we have ( u u n V 1 + b ) inf u w n γ n w V n W n For any u V we define some projection I n : V W n, that we can estimate Trivially, we have inf u w n w V u I n u V (33) n W n For example, for bounded functions u, e g, for u H (Ω) we can project onto the space of linear finite elements by I n u = u(p )b P (x), P N (M) which is an evaluation of the values on the nodes The interpolation operator of Raviart-Thomas is defined as (I n v)(x) := N l j (v)b j (x) j=1 where l j : V R/C are functionals, called global degrees of freedom Number of global degrees of freedom equals the number of basis functions Lemma 36 (Bramble-Hilbert lemma) If Ω R d is a bounded Lipschitz-domain and m N, then γ = γ(m, Ω) > : inf v p H p P m (Ω) γ v H m (Ω) v H m (Ω) m(ω) Generalization of Poincaré s inequality (Lemma 6) and a best approximation of polynomials in Sobolev spaces which is similar to the pointwise statement of Taylor s theorem The constant γ depends on the domain Using reference element is beneficial as the constant is then always the same The global degrees of freedom, e g, point evaluations, edge integrals or cell integrals, define on local degrees of freedom and the local interpolation operator Îû = (I n u)(x(ξ)), which acts on the pull-back û(ξ) = u(φ (ξ)) of u Estimate on the reference element Using the triangle inequality, the fact that there exists a constant C(t, m, ), 1 t m + 1 if d = 1 and t m + 1 if d =, 3, such that Îû H t ( ) C(t, m, ) û Ht ( ) for all û H t ( ), which follows from the continuous embedding of H t ( ) in C ( ) and, finally, using the Bramble-Hilbert lemma we obtain an estimate for the interpolation error of Î Let us measure the error in the H r -norm with r t m + 1 (t 1 for d = 1 and t for d =, 3) Then û Îû Hr ( ) û Îû Ht ( ) = inf (û p) Î(û p) p P m( ) Ht ( ) ( û inf p H t( p P m( ) ) + Î(û p) )) Ht( (34) (1 + C(t, m, )) inf p P m( ) û p Ht ( ) γ(m, )(1 + C(t, m, }{{ )) û } H t ( ) =C(t,m), since is fixed 33

10 Transformation techniques we transform Sobolev norms To apply the Bramble-Hilbert lemma on and transforming back to Lemma 37 If Φ : is an affine mapping ξ F ξ + τ, then, for all m N, ( ) m + d û Hm ( ) d m F m det(f ) / u d H m () u H m (), ( ) m + d u H m () d m F d m det(f ) 1 / û H m ( ) u H m ( ) with F denoting the matrix norm of F associated with the Euclidean vector norm Estimate on a cell With the transformation techniques presented above we can relate the interpolation error on with that one of the pull-back û u I n u r H r () = u I n u H l () l= r ( ) l + d d l F d l det(f ) û Î n u }{{} l= Îû H l ( ) ( ) r + d C(r) d r F d r det(f ) û Îû H r ( ) Using the estimate (34) on the reference element we get u I n u ( ) r + d C(r, t, m) Hr d r F () d r det(f ) 1 / û Ht ( ), and transformed back to u In u ( )( ) r + d t + d C(r, t, m) H d r+t F r () d d r F t u H t (), (35) }{{} C(d,t,r,m) with r t m + 1, t The estimate depends on the size and shape of the triangle through F Note that convergence in terms of h is only expected if the solution has higher regularity, i e, u H () if we measure in H 1 () Definition 38 Given a cell of a mesh M we define its diameter h := sup{ x y, x, y }, and the maximum radius of an inscribed ball r := sup{r > : x : x y < r y } The ratio ρ = h /r is called the shape regularity measure of r l α r h Figure 37: Diameter h and maximum radius r of an inscribed ball for a triangular cell Figure 38: Angle condition for shape regularity of a triangle The shape regularity measure of a simplex can be calculated from bounds for the smallest and largest angles enclosed by edge/face normals For triangles with the smallest angle α > we have sin( α /) ρ sin( α /) 34

11 Estimates for affine elements maps ρ 1 d M µ d M with the meshwidth h d M det(f ) h d M, F h M, F ρm µ M h M, (36) h M := max{h, M}, the shape regularity measure ρ M := max{ρ, M}, and the quasi-uniformity measure µ M := max{h /h,, M} = max{h, M} min{h, M} Interpolation error estimate in the mesh The interpolation error can be decomposed into the contributions from all triangles of M, using (35) and (36), and summing the errors from each triangle we get Theorem 39 Let I n stand for the finite element interpolation operator belonging to the finite element space S m (M) on a simplicial mesh M Then, for t m + 1, r t γ = γ(t, r, m, ρ M, µ M ) : u I n u H r (Ω) γ ht r M u H t (Ω) u H t (Ω) Discretization error in the mesh From (3) and (33), i e, from the fact that Discretization error best-approximation error interpolation error and Theorem 39 we can deduce that the discretization error satisfies u u n H 1 (Ω) γ ht M u H t (Ω) for t m + 1 and u H t (Ω), (37) where γ = γ(ω, γ n, b, ρ M, µ M, m, t) Discussion To expect convergence we need u to be of higher regularity For certain problems higher regularity u H r+ (Ω) is guaranteed by elliptic shift theorems if f H r (Ω), and the boundary and the boundary data are smooth The bound (37) can be interpreted as asymptotic a-priori error estimate u u n H 1 (Ω) = O(ht M n ) for n, where M n, n N, is a sequence of simplicial, uniformly shape-regular (ρ Mn < ρ max for all n N) meshes of Ω, that become infinitely fine (h Mn as n ) In other words, we obtain a convergence with the meshwidth of order t 1, which is equal to the polynomial degree if the solution u is sufficiently regular If Ω is convex or has C 1 -boundary, and a C 1 (Ω), then u u n L (Ω) γh M u u n H1 (Ω) with γ = γ(ω, a, m, ρ M, µ M ) > In other words, the weaker the norm of the discretization error the faster it converges to zero as h M A more sophisticated best approximation estimates for increasing polynomial degree, i e, m, t 1, reads inf u v n H v n S m (M) 1 (Ω) γ(ρ M, µ M ) ( ) min{m+1,t} hm u m Ht(Ω) (38) which is an a-priori error estimates for the p-version of H 1 -conforming finite elements 35