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The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli Visiting Professor, Wuppertal University January 6, 2016

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli Collaborations E. Georgoulis, J. Liesen, D. Loghin, A.Miedlar, E. Noulard, D. Orban, A. Russo, Z. Strakos, and A. Wathen 2 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli Problem H 1 0 ( ! ) the standard Sobolev space of functions with zero trace on @! . Let Ω be a bounded open polyhedral domain in R d , d = 2 , 3 and let @ Ω denote its boundary. We consider the second order equation ( | ) � r · ( a r u ) = f in Ω , where a 2 [ L ∞ ( Ω )] d × d is a positive definite tensor and f 2 L 2 ( Ω ). For simplicity of the presentation, we impose homogeneous Dirichlet boundary condition u = 0 on @ Ω , although this appears not to be an essential restriction. We shall denote by || · || a := kp a r ( · ) k the, so-called, energy norm. 3 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli FEM Let T be a conforming subdivision of Ω into disjoint simplicial elements  2 T . We assume that the subdivision T is shape-regular and that it is constructed via a ffi ne mappings F  , where F  : ˆ  !  , with non-singular Jacobian, where ˆ  is the reference simplex. For a nonnegative integer r , we denote by P r (ˆ  ), the set of all polynomials of total degree at most r , defined on ˆ  . We consider the finite element space V := { V 2 H 1 0 ( Ω ) : V |  � F  2 P r (ˆ  ) ,  2 T } . 4 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli FEM By Γ we denote the union of all ( d � 1)-dimensional element faces associated with the subdivision T (including the boundary). Further we decompose Γ into two disjoint subsets Γ = @ Ω [ Γ int , where Γ int := Γ \ @ Ω . We define h  := ( µ d (  )) 1 / d ,  2 T , where µ d is the d -dimensional Lebesgue measure. Also, for two (generic) elements  + ,  − sharing a face e := @ + \ @ − ⇢ Γ int we define h e := µ d − 1 ( e ). We collect these quantities into the element-wise constant function h : Ω ! R , with h |  = h  ,  2 T and h | e = h e , e 2 Γ . The families of meshes constructed by the algorithms presented in this work will be conforming and shape-regular. 4 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli FEM The finite element method reads: ( F ) find U 2 V such that a ( U , V ) = l ( V ) 8 V 2 V , where the bilinear form a : H 1 0 ( Ω ) ⇥ H 1 0 ( Ω ) ! R and the linear form l : H 1 0 ( Ω ) ! R are given by Z Z a ( w , v ) := a r w · r v d x and l ( v ) := fv d x , Ω Ω respectively, for w , v 2 H 1 0 ( Ω ). 4 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli FEM Let now { � i } 1 ≤ i ≤ N denote a set of basis functions for V so that N X U = u i � i , i =1 and let A ij = a ( � j , � i ) , b k = l ( � k ) , i , j , k = 1 , · · · , N . With this notation, the linear system corresponding to is Au = b , R N × N is the sti ff ness matrix corresponding to a set of where A 2 I basis functions { � i } 1 ≤ i ≤ N . 4 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM For every face e 2 Γ int , we define the jump across e of a scalar function w , defined in an open neighbourhood of e , by ⌘ � [ w ]( x ) = lim w ( x � t n e ) � w ( x + t n e ) , t → 0 for x 2 e , where n e denotes a normal vector to e . (Note that the jump is only uniquely defined up to a sign, which is unimportant for the discussion below.) For any subset M ⇢ T (i.e., M is a collection of elements of T ), we define the local estimator by ⌘⌘ 1 / 2 ⇣ X ⇣ h 2  k f + r · ( a r U ) k 2 h e k [ a r U · n e ] k 2 X ⌘ T ( U , M ) :=  + . e  ∈ M e ∈ Γ int ∩ @ 5 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM Algorithm 1. AFEM algorithm Set parameter 0 < ✓  1. Set m = 0. While convergence criterion not satisfied 1. Solve exactly ( F ) to obtain U e m (the exact solution). 2. Compute local estimators ⌘ T m ( U e m ,  ),  2 T m . 3. Mark elements M m for refinement in T m using (D¨ orfler marking) ⌘ 2 T m ( U e m , M m ) � ✓ ⌘ 2 T m ( U e m , T m ) . 4. Refine M m to obtain new mesh T m +1 . Set m m + 1. End 5 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM SOLVE ! ESTIMATE ! MARK ! REFINE Theorem There exist constants ⇠ > 0 and 0 < ↵ < 1 such that ⇣ ⌘ k u � U e m +1 k 2 a + ⇠⌘ 2 T m +1 ( U e k u � U e m k 2 a + ⇠⌘ 2 T m ( U e m +1 , T m +1 )  ↵ m , T m ) . (Cascon, Kreuzer, Nochetto, and Siebert SINUM 2008) 5 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM SOLVE ! ESTIMATE ! MARK ! REFINE 6 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM APPROXIMATE ! ESTIMATE ! MARK ! REFINE 6 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM Algorithm 2. Inexact AFEM Set parameters 0 < ✓  1, µ and ⌫ . Initialise ˜ U 0 . Set m = 1. While convergence criterion not satisfied 1. Solve inexactly ( F ) to obtain ˜ U m so that U m − 1 � U m − 1 k 2 + µ k ˜ U m � U m k 2  ⌫⌘ 2 k ˜ m − 1 ( ˜ U m − 1 ) , for some values µ and ⌫ is satisfied. T m ( ˜ U m ,  ),  2 ˜ 2. Compute local estimators ⌘ ˜ T m . M m for refinement in ˜ ˜ 3. Mark elements T m using ⌘ 2 U m , M m ) � ✓ ⌘ 2 T m ( ˜ T m ( ˜ U m , T m ) . M m to obtain new mesh ˜ ˜ 4. Refine T m +1 . Set m m + 1. End 6 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM Theorem Let u , ˜ U m and ˜ U m +1 , m � 1 (approximations of U m and U m +1 solutions on ˜ T m and ˜ T m +1 ) , be such that || ˜ U m � U m || 2 a + µ || ˜ U m +1 � U m +1 || 2 a  ⌫⌘ 2 m ( ˜ U m ) , with µ := 1 + ⇠ C 1 (1 + � − 1 ) � , ⌫ := 1 + 2 C 2 C 1 ) , � � 1 + 2 C 2 ) ✏⇠ ✏ � − 1 , and � , � , � and ✏ 2 C 1 (1 + � )(1 + � − 1 ) � where 0 < ✏ < 1 , ⇠ := are chosen small enough, so that (1 � ⌧✓ )(1 + � ) + 2 ✏ C 2 + � < 1 . Then, there exist a constant 0 < ↵ < 1 , depending only on the shape regularity of ˜ T 1 and on the marking parameter ✓ , such that || u � ˜ U m +1 || 2 a + ⇠⌘ 2 m +1 ( ˜ || u � ˜ U m || 2 a + ⇠⌘ 2 m ( ˜ � � U m +1 )  ↵ U m ) . (A., Georgoulis, and Loghin SISC, 2013) 7 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM U m � U m || 2 U m +1 � U m +1 || 2 a  ⌫⌘ 2 || ˜ a + µ || ˜ m ( ˜ U m ) , 8 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM U m � U m || 2 || ˜ a 8 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli AFEM 7� ! iAFEM A m u m = b m . R N m × N m with { N m } m an increasing sequence. The matrices A m 2 I k U m � U k m k a = k u m � u k m k A m where h x , y i A := x T Ay , x , y 2 I R N , A 2 R N × N , denotes the standard inner product weighted by A in R N , with the p corresponding norm k x k A := h x , x i A . 8 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli Krylov + CG I We need a method that includes an energy-norm estimator (possibly an upper bound) of the errors! I It would be desirable to have a monotonic sequence! 9 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli Krylov + CG R N m be the k -th CG iterate at step m of the adaptive Let u k m 2 I algorithm and by U k m the corresponding function in ˜ V m . We denote the residual by r k m := b m � A m u k m and note that the energy norm of the error can be expressed as a dual norm of the residual: k U m � U k m k a = k u m � u k m k A m = k r k m k A � 1 m , 9 / 33

The use of stopping criteria for iterative Krylov methods in designing adaptive methods for PDEs Mario Arioli Krylov + CG It is well-known that the Conjugate Gradient method minimises the energy norm of the error, namely u k m = arg min k u m � u k A m , u ∈ K k ( r 0 m , A m ) n o where K k ( r 0 r 0 m , A m r 0 m , · · · , A k − 1 r 0 m , A m ) := is the Krylov m m subspace of dimension k . Thus, the energy norm of the error decreases monotonically and the criterion needed will be satisfied for m with k > k ∗ for some k ∗ . all U k In addition, there are various established numerical techniques that provide bounds or estimates for the energy norm of the error at each step. 9 / 33