A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions

Transcription

1 Lin Lin A Posteriori DG using Non-Polynomial Basis 1 A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions Lin Lin Department of Mathematics, UC Berkeley; Computational Research Division, LBNL Joint work with Benjamin Stamm Dimension Reduction: Mathematical Methods and Applications, Penn State University, March, 2015 Supported by DOE SciDAC Program and CAMERA Program

2 Lin Lin A Posteriori DG using Non-Polynomial Basis 2 Outline Introduction: Adaptive local basis functions Computable upper bound for Poisson s equation Computable upper / lower bound for indefinite equations Numerical examples Conclusion and future work

3 Lin Lin A Posteriori DG using Non-Polynomial Basis 3 Motivation Spatially inhomogeneous quantum systems Ω

4 Lin Lin A Posteriori DG using Non-Polynomial Basis 4 Kohn-Sham density functional theory HH ρρ ψψ ii xx = 1 2 Δ + vv eeeeee xx + ddxx ρρ xx NN/2 xx xx + VV xxxx ρρ ψψ ii xx = εε ii ψψ ii xx ρρ xx = 2 ψψ ii xx 2, dddd ψψ ii xx ψψ jj xx = δδ iiii, εε 1 εε 2 ii=1 Efficient: Always solve an equation in RR 3, regardless of the number of electrons NN. Accurate: Exact ground state energy for exact VV xxxx [ρρ], [Hohenberg-Kohn,1964], [Kohn-Sham, 1965] Best compromise between efficiency and accuracy. Most widely used electronic structure theory for condensed matter systems and molecules Nobel Prize in Chemistry, 1998

5 Lin Lin A Posteriori DG using Non-Polynomial Basis 5 Discretization cost Basis Example DOF / atom Construction Uniform basis Planewave Finite difference Finite element 500~10000 or more Simple and systematic Quantum chemistry basis Gaussian orbitals Atomic orbitals 4~100 Fine tuning Non-systematic convergence Q: Combine the advantage of both?

6 Lin Lin A Posteriori DG using Non-Polynomial Basis 6 Adaptive local basis functions Idea: Use local eigenfunctions as basis functions How to patch the basis functions together?

7 Lin Lin A Posteriori DG using Non-Polynomial Basis 7 Discontinuous Galerkin method Kohn-Sham New terms [LL-Lu-Ying-E, J. Comput. Phys. 231, 2140 (2012)] Interior penalty method [Arnold, 1982]

8 Lin Lin A Posteriori DG using Non-Polynomial Basis 8 Why a posteriori error estimator Measuring the accuracy of eigenvalues and densities without performing an expensive converged calculation, or benchmarking with another code. Optimal allocation of basis functions for inhomogeneous systems.

9 Lin Lin A Posteriori DG using Non-Polynomial Basis 9 Residual based a posteriori error estimator Vast literature for second order PDE and eigenvalue problems Polynomial basis functions, finite element: [Verfürth,1996] [Larson, 2000] [Durán-Padra-Rodríguez, 2003] [Chen-He-Zhou, 2011]... Polynomial basis functions, discontinuous Galerkin: [Karakashian-Pascal, 2003], [Houston-Schötzau-Wihler, 2007], [Schötzau-Zhu, 2009], [Giani-Hall, 2012]...

10 Lin Lin A Posteriori DG using Non-Polynomial Basis 10 Difficulty A posteriori error analysis relies on the detailed knowledge of asymptotic approximation properties of the basis set Difficult for equation-aware basis functions Adaptive local basis functions Heterogeneous multiscale method (HMM) [E-Engquist 2003] Multiscale finite element [Hou-Wu 1997] Multiscale discontinuous Galerkin [Wang-Guzmán- Shu, 2011] etc

11 Lin Lin A Posteriori DG using Non-Polynomial Basis 11 Outline Introduction: Adaptive local basis functions Computable upper bound for Poisson s equation Computable upper / lower bound for indefinite equations Numerical examples Conclusion and future work

12 Lin Lin A Posteriori DG using Non-Polynomial Basis 12 Model problem Discontinuous space (broken Sobolev space) κκ FF Ω VV NN HH 2 (KK) Piecewise constant function belongs to VV NN

13 Lin Lin A Posteriori DG using Non-Polynomial Basis 13 DG discretization Bilinear form (θθ = 1 corresponds to the symmetric form) Define the inner products Average and jump operators

14 Lin Lin A Posteriori DG using Non-Polynomial Basis 14 Error quantification DG approximation Error in the broken energy norm Goal: Find a sharp upper bound for

15 Lin Lin A Posteriori DG using Non-Polynomial Basis 15 Upper bound of error Theorem ([LL-Stamm 2015]). Let uu HH # 1 Ω HH 2 (KK) be the true solution and uu NN VV NN the DG-approximation. Then where Residual Jump of gradient Jump of function The key is to find the dependence of aa κκ, bb κκ, cc κκ w.r.t. VV NN.

16 Lin Lin A Posteriori DG using Non-Polynomial Basis 16 Projection operator LL 2 κκ -projection operator Inner product Similar to HH 1 (κκ) norm Projection operator onto basis space Therefore

17 Lin Lin A Posteriori DG using Non-Polynomial Basis 17 Estimating constants Define is in the sense of the inner product,,κκ Lemma. Let κκ KK, vv HH 1 κκ. Then Proof: Similar for bb kk

18 Lin Lin A Posteriori DG using Non-Polynomial Basis 18 Numerical procedure for computing the constants Basic idea: estimate the constants by iteratively solving generalized eigenvalue problems on an infinite dimensional space 1D demonstration, generalizable to any d-dimension. Consider κκ = 0, h, spectral discretization with Legendre- Gauss-Lobatto (LGL) quadrature: 0 yy jj h NN gg NN gg Integration points yy jj jj=1, integration weights ωω jj jj=1

19 Lin Lin A Posteriori DG using Non-Polynomial Basis 19 Numerical representation of inner product LGL grid points defines associated Lagrange polynomials of degree NN gg 1 Approximate any vv HH 1 κκ Define Inner product

20 Lin Lin A Posteriori DG using Non-Polynomial Basis 20 Numerical representation of inner product,κκ requires differentiation matrix Differentiation becomes matrix-vector multiplication

21 Lin Lin A Posteriori DG using Non-Polynomial Basis 21 Numerical representation of inner product Projection onto constant In sum

22 Lin Lin A Posteriori DG using Non-Polynomial Basis 22 Estimating aa kk Here Handling the orthogonal constraint by projection QQ = II Π NN κκ

23 Lin Lin A Posteriori DG using Non-Polynomial Basis 23 Estimating aa kk This is a generalized eigenvalue problem Solve with iterative method, such as the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method [Knyazev 2001] Only require matrix-vector multiplication.

24 Lin Lin A Posteriori DG using Non-Polynomial Basis 24 Estimating bb kk How to estimate uu, vv. Importance of Lobatto grid Here MM bb = WW

25 Lin Lin A Posteriori DG using Non-Polynomial Basis 25 Generalize to high dimensions Tensor product LGL grid Tensor product Lagrange polynomials ll

26 Lin Lin A Posteriori DG using Non-Polynomial Basis 26 Compare with asymptotic results for polynomial basis functions For polynomial basis functions [e.g. Houston-Schötzau- Wihler, 2007] h = 1

27 Lin Lin A Posteriori DG using Non-Polynomial Basis 27 Penalty parameter Parameter {γγ κκ } Large enough for coercivity of the bilinear form magic parameter in interior penalty method [Arnold 1982] Define Lemma. If γγ κκ 1 2 coercive 1 + θθ 2 dd κκ 2, then the bilinear form is Automatic guarantee of stability

28 Lin Lin A Posteriori DG using Non-Polynomial Basis 28 Penalty parameter Computation of dd κκ through eigenvalue problem By setting vv NN = Φcc, span Φ = VV NN κκ. Can be solved with direct method

29 Lin Lin A Posteriori DG using Non-Polynomial Basis 29 Upper bound estimator The last constant dd κκ uu (uu NN ) involves the true solution uu and therefore is the only constant that cannot be explicitly computed. However, numerical result shows that dd uu κκ uu NN dd κκ is a good approximation.

30 Lin Lin A Posteriori DG using Non-Polynomial Basis 30 Outline Introduction: Adaptive local basis functions Computable upper bound for Poisson s equation Computable upper / lower bound for indefinite equations Numerical examples Conclusion and future work

31 Lin Lin A Posteriori DG using Non-Polynomial Basis 31 Model problem Indefinite equation VV LL Ω and Δ + VV has no zero eigenvalue. Bilinear form DG approximation

32 Lin Lin A Posteriori DG using Non-Polynomial Basis 32 Computable upper bound Energy norm Theorem ([LL-Stamm 2015]). Let uu HH # 1 Ω HH 2 (KK) be the true solution and uu NN VV NN the DG-approximation. Then

33 Lin Lin A Posteriori DG using Non-Polynomial Basis 33 Computable lower bound Theorem ([LL-Stamm 2015]). Let uu HH # 1 Ω HH 2 (KK) be the true solution and uu NN VV NN the DG-approximation. Then where All constants other than dd κκ uu are computable

34 Lin Lin A Posteriori DG using Non-Polynomial Basis 34 Computable lower bound Bubble function bb κκ For instance, bb κκ xx = 4 xx 1 xx, κκ = 1 Lemma. where

35 Lin Lin A Posteriori DG using Non-Polynomial Basis 35 Outline Introduction: Adaptive local basis functions Computable upper bound for Poisson s equation Computable upper / lower bound for indefinite equations Numerical examples Conclusion and future work

36 Lin Lin A Posteriori DG using Non-Polynomial Basis 36 1D Poisson equation Δuu xx = sin 6xx Adaptive local basis functions with 11 basis per element.

37 Lin Lin A Posteriori DG using Non-Polynomial Basis 37 Effectiveness of upper/lower estimtaor Measure local effectiveness (CC ηη 1, CC ξξ 1)

38 Lin Lin A Posteriori DG using Non-Polynomial Basis 38 1D indefinite Δuu xx + VV xx uu(xx) = sin 6xx Adaptive local basis functions with 11 basis per element.

39 Lin Lin A Posteriori DG using Non-Polynomial Basis 39 Effectiveness of upper/lower estimtaor

40 Lin Lin A Posteriori DG using Non-Polynomial Basis 40 2D Helmholtz Δuu + VVVV = ff, VV = 16.5, ff xx, yy = ee 2 xx ππ 2 2 yy ππ 2 Adaptive local basis functions with 31 basis per element.

41 Lin Lin A Posteriori DG using Non-Polynomial Basis 41 Effectiveness for upper/lower bound

42 Lin Lin A Posteriori DG using Non-Polynomial Basis 42 Validate the approximation for dd κκ uu Note that Although dd κκ uu is not known, it is only sufficient to have dd κκ uu dd κκ or dd κκ uu bb κκ γγ κκ 1D:

43 Lin Lin A Posteriori DG using Non-Polynomial Basis 43 Validate the approximation for dd κκ uu 2D indefinite

44 Lin Lin A Posteriori DG using Non-Polynomial Basis 44 Conclusion Systematic derivation of a posteriori error estimation for general non-polynomial basis function Explicitly computable constants for upper/lower estimator. The only one non-computable constant can be reasonably estimated by known ones.

45 Lin Lin A Posteriori DG using Non-Polynomial Basis 45 Future work Eigenvalue problem Nonlinearity, atomic force, linear response properties Implementation in DGDFT Other basis functions, including MsFEM, HMM, MsDG etc. Ref: LL and B. Stamm, A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part I: Second order linear PDE, arxiv: Thank you for your attention!