Efficient Multigrid based solvers for Isogeometric Analysis

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1 Efficient Multigrid based solvers for Isogeometric Analysis R. Tielen, M. Möller and C. Vuik Delft Institute of Applied Mathematics (DIAM) Numerical Analysis / 22

2 Isogeometric Analysis (IgA) Extension of the Finite Element Method (FEM) Same basis functions (B-Splines) are used for approximate geometry Ωh and solution uh Global mapping from Ωh to parametric domain Ω h Description of the geometry that is highly accurate ( Ω = Ωh ) throughout all computation steps Figure: Poisson problem solved by FEM (left) and IgA (right). 2 / 22

3 Construction of B-spline basis functions A knot vector is a sequence of non-decreasing points ξi R with the following structure: Ξ = (ξ, ξ2,..., ξi,..., ξn+p, ξn+p+ ) where n is the number of B-spline basis functions p is the degree of the basis functions Ξ is called open and uniform if: The first and last knots are repeated p + times All ξp+,..., ξn+ are equally spaced 3 / 22

4 Construction of B-spline basis functions Cox-de Boor recursion formula ( if ξi ξ < ξi+ φi, (ξ) = otherwise. φi,p (ξ) = ξi+p+ ξ ξ ξi φi,p (ξ) + φi+,p (ξ) ξi+p ξi ξi+p+ ξi+ for p, where ξ [ξ, ξn+p+ ]. 4 / 22

5 Examples of B-spline basis functions (p = ) basis functions for Ξ = {,, 2, 3} 5 / 22

6 Examples of B-spline basis functions (p = ) basis functions for Ξ = {,, 2, 3} 5 / 22

7 Examples of B-spline basis functions (p = ) basis functions for Ξ = {,, 2, 3} 5 / 22

8 Examples of B-spline basis functions (p = ) Linear basis functions for Ξ = {,,, 2, 3, 3} 6 / 22

9 Examples of B-spline basis functions (p = 2) Quadratic basis functions for Ξ = {,,,, 2, 3, 3, 3} 7 / 22

10 Properties of B-spline basis functions Compact support Sparse system matrices Mass matrix positive Strictly positive Partition of unity Direct mass lumping / 22

11 B-spline basis functions in 2D Extension 2D Tensor product of the D B-spline basis functions 9 / 22

12 Need for efficient solvers Condition number of stiffness/mass matrix κ(ah,p ) scales exponentially with approximation order p κ(ah,p ) p 5 6 / 22

13 Observation The linear system Ah,p xh,p = bh,p reduces to standard FEM for p = ; becomes more difficult to solve for increasing p. Efficient solvers for high-order B-spline-based discretizations are needed Solution strategy Use the error of low-order discretizations to update the solution of high-order discretizations p-multigrid / 22

14 p-multigrid Hierarchy of discretizations with different orders k Low-order error is used to update high-order solution Smoothing steps are applied at each k-level ( ) Pr ict str Re olo ng ate k=3 Pr olo ng a ict str Re k= te k=2 Solve residual equation 2 / 22

15 Prolongation/Restriction Restrict residual rk from level k to level k : k Ikk := (Mk k ) Mk Prolongate error ek from level k to level k: k Ik := (Mkk ) Mkk Where: (Mlk )(i,j) := R Ω h φki (ξ) φlj (ξ) c(ξ) dω Mkk is in practice replaced by its lumped counterpart 3 / 22

16 V-cycle p-multigrid Solution procedure () Start with initial guess uh,p (n) Obtain correction e h,p with single V-cycle Solution update: (n+) uh,p (n) (n) uh,p + e h,p Stopping criterion: (n) rh,p () < rh,p 4 / 22

17 Numerical Results Consider u = f on Ω u = uexact on Ω where uexact (x, y ) = (x 2 + y 2 )(x 2 + y 2 4)xy 2 5 / 22

18 p-multigrid as a solver SOR (τ = 43 ) for pre/post-smoothing (ν = 4) Conjugate Gradient at level k = ( = 4 ) 2 p = 2, h = 2 4 p = 2, h = 2 5 p=2 Residual V-cycles h-refinement level V-cycles / 22

19 p-multigrid as a solver SOR (τ = 43 ) for pre/post-smoothing Conjugate Gradient at level k = ( = 4 ) 5 p=2 p=3 p=4 Residual V-cycles p = 2, h = 2 5 p = 3, h = 2 4 p = 4, h = h-refinement level V-cycles / 22

20 p-multigrid as a solver 2 p=2 p=3 p=4 L2 error h-refinement level 6 Optimal spatial convergence O(hp+ ) 7 / 22

21 p-multigrid as a preconditioner Conjugate Gradient as outer solver ( = 8 ) V-cycle as preconditioner in every iteration 3 2 p=2 p=3 p=4 4 2 L2 error Iterations p=2 p=3 p= h-refinement level h-refinement level 6 8 / 22

22 Observations Numerical results indicate: Number of V-cycles/iterations is relatively low 3 Number of V-cycles/iterations is h-independent 3 Optimal O(hp+ ) spatial convergence is achieved 3 Number of V-cycles/iterations is p-dependent 7 9 / 22

23 Spectral analysis Error reduction factors: r S (v) = S(v) v r CGC (v) = CGC (v) v where S( ) and CGC ( ) denote a smoothing step and coarse grid correction applied on v, respectively. Here (vi ) are the generalized eigenvectors which satisfy: Ah,p vi = λi MC h,p vi, i =,..., Ndof 2 / 22

24 Reduction Reduction Spectral analysis.2.2 CGC Smoother CGC Smoother Eigenvector 2 3 Eigenvector Figure: Reduction factors (v i ) for p = 2 (left) and p = 3 (right). 2 / 22

25 Forthcoming Work Obtain p-independence by alternative smoothers ( ) Explore flexibility of coarsening in both h and p I Ndof at coarsest level is relatively high ( ) C. Hofreither and S. Takacs. Robust Multigrid for IgA Based on Stable Splittings of Spline Spaces SIAM Journal on Numerical Analysis, 55(4): , / 22

Algebraic Flux Correction Schemes for High-Order B-Spline Based Finite Element Approximations

Algebraic Flux Correction Schemes for High-Order B-Spline Based Finite Element Approximations Numerical Analysis group Matthias Möller m.moller@tudelft.nl 7th DIAM onderwijs- en onderzoeksdag, 5th November