Superconvergence Points in Finite Element Method

1. Superconvergence Points in Finite Element Method Zhimin Zhang & Runchang Lin Department of Mathematics Wayne State University Detroit, MI 48202 http://www.math.wayne.edu/~zzhang Research is partially supported by the NSF grants: DMS-0612908 and DMS-0311807

2. Finite Element Superconvergence What is Finite Element Superconvergence? The rate of convergence of the finite element solution and / or its gradient at some special points exceeds the possible global rate. • The global optimal rate of convergence • Superconvergence • In this talk, we mainly consider: Natural superconvergence points

3. Previous works (2) Finite Element Superconvergence • A Breakthrough in 90s • Computer-Based Proof of Superconvergence (Babuška et al.) • Assumptions: • No round off errors; • Mesh is locally translation-invariant; • Pollution error is under control.

4. Previous works (3) Previous works (3) Finite Element Superconvergence • A Breakthrough in 90s • Theorem 1. Babuška, Strouboulis, Upadhyay, Gangaraj, 1996 • where , is a periodic polynomial of degreen + 1. • “A superconvergence point exists if and only if it is a common zero point for a finite number of piecewise-polynomial functions.” • Remark: Polynomials are computed numerically.

5. What is left Finite Element Superconvergence Previous works (6) Issues not Solved by Computer-Based Proof • Computer findings need to be theoretically verified • Function value superconvergence points • Superconvergence points for equilateral triangle • Superconvergence points for for 3-D elements

6. Finite Element Superconvergence Observation • Superconvergence points are very sensitive to: • Solution properties of PDEs. • Finite element spaces. • Geometric pattern of meshes. • Results are valid for (with smooth data) • The Poisson equation • 2nd order elliptic equation • Linear elasticity equation • The Reissner-Mindlin plate model • (locking free elements) • Convection-diffusion equation • (when layers are resolved) • ... ...

7. Finite Element Superconvergence • Key: Characteristic of the periodic polynomial space • where • and An Analytic Approach

8. Summary (1) Finite Element Superconvergence Previous works (6) What is left Summary of Our Results • Validate the accuracy of Computer-based Proof • Function Value Superconvergent Points • Triangular elements: 5 different patterns include equilateral • Rectangular elements: Intermediate and serendipity families • 3-D tetrahedral, pentahedral, Hexahedral elements • Laplace equation: New results for non-symmetry points

9. Finite Element Superconvergence • Gaussian points: Zeros of the Legendre polynomials. • Complete polynomial space: • Tensor-product space: • Intermediate family of type I: • Intermediate family of type II: • Serendipity family: Notation and Terminology

11. where is the Legendre polynomial of degree k.

12. Finite Element Superconvergence • Tensor-product space, intermediate families of type I & II, serendipity family ofr = 1,2 • One direction derivative: Along the Gaussian lines. • Stress points: At the Gaussian points. • Serendipity family forr  3 • One direction derivative • r = 3:Center line = 0, • r = 4, 6, 8, …:None • r = 5, 7, 9, …:Element center and two edge center • Stress points • r = 1, 3, 5, …:Element center; r = 2, 4, 6, …:None Superconvergence Points: Rectangular Element

13. Finite Element Superconvergence 1. Space between tensor-product and intermediate family Superconvergence Points: Brick Elements • 1.1. Function value superconvergence results • Tensor product of Lobatto points (zeros of φr+1) • 1.2. Derivative superconvergence results • One direction derivative: On the Gaussian surfaces. • Two direction derivative: Along the Gaussian lines. •  Stress points: At the Gaussian points. The results for the Poisson and Laplace equations are the same!

14. Finite Element Superconvergence Superconvergence Points: Brick Elements 2. Serendipity family Vr, where P(r+1)\Vr = Span{ξiηjζk| i+j+k = r+1, i, j, k 1} • 2.1. Function value superconvergence results: • r is even: mesh symmetry points • r (>3) is odd: no superconvergence point • r = 3 • Poisson equation -- no superconvergence point • Laplace equation -- 32 points: • where

15. Finite Element Superconvergence Superconvergence Points: Brick Elements • 2.2. ξ-derivative superconvergence results: • r = 1 and 2: the Gaussian plane(s) Pr(ξ) = 0 • r = 3: • Poisson equation: the plane ξ = 0 and eight points • Laplace equation: the plane ξ = 0 and curves • r = 4 and 6: no superconvergence point • r= 5: nine points

16. Mesh Patterns • Uniform Triangular Mesh Patterns Regular Pattern Chevron Pattern Criss-Cross Pattern Union Jack Pattern

17. Finite Element Superconvergence • Four mesh patterns; One direction derivative • Regular • r = 1, 3, 5, …: Horizontal edge center • r = 2: Two Gaussian points on the horizontal edge • r = 4, 6, …:None • Chevron • r = 1, 3, 5, …: Horizontal edge center • r = 2, 4, 6, …:None • Union-Jack: None • Criss-Cross • T1: The same as Regular • T2: None Superconvergence Points: Triangular Elements

19. Summary (2) Finite Element Superconvergence Previous works (6) What is left • Triangular Elements • Derivative Superconvergent Points: • y-derivative, Chevron pattern: (not reported in the literature) • Linear element: Midpoint of vertical side, • a non-symmetry point! * • Higher order elements: No superconvergent points. • Verifying computer findings have 9 digits accuracy, except a pair of points in cubic element under Criss-Cross pattern (7 digits);

20. Regular Pattern Finite Element Superconvergence in Triangular Elements Regular Pattern

21. Superconvergence for Laplace Equation Finite Element Superconvergence in Triangular Elements • Superconvergence for Laplace Equation • To determine and , adding basis functions of • corresponding to terms in Re(zn+1) and Im(zn+1).

22. Function Value Superconvergent Points for Laplace Equation (1) Finite Element Superconvergence in Triangular Elements • Function Value Superconvergence for Laplace Equation • n=1 (linear FE), function value superconvergent points in T1 are solutions of • which are • n=2 (quadratic FE), function value superconvergent points in T1 are solutions of • which are vertices, midpoints of edges, and

23. Function Value Superconvergent Points for Laplace Equation (2) Finite Element Superconvergence in Triangular Elements • Function Value Superconvergence for Laplace Equation

24. x-Derivative Superconvergent Points for Laplace Equation Finite Element Superconvergence in Triangular Elements • x-Derivative Superconvergence for Laplace Equation

25. One Monster Chevron Pattern Finite Element Superconvergence in Triangular Elements One Monster!

26. Other Patterns (1) Finite Elemetn Superconvergence in Triangular Elements • Function Value Superconvergence for Other Patterns • Poisson Equation (Conclusive!) • Odd FEs: No superconvergent points. • Even FEs: Only symmetry points.

27. Other Patterns (2) Finite Element Superconvergence in Triangular Elements • Function Value Superconvergence • Laplace Equation - Chevron, Union Jack • Linear FE: Only vertices (Different from the Regular pattern). • Higher order FEs: From Regular pattern by symmetry. Example n = 8.

28. Other Patterns (3) Finite Element Superconvergence in Triangular Elements • Function Value Superconvergence • Laplace Equation - Criss Cross

29. Finite Element Superconvergence Equilateral Triangular Elements

30. Finite Element Superconvergence ξ-derivative Superconvergence Points for Equilateral Triangular Elements

31. Finite Element Superconvergence Pentahedral Elements 1. Superconvergence results of tensor-product elements: 1.1. Function value : tensor-product of triangular function value superconvergence points in ξη plane and 1D Lobatto points in ζ direction 1.2. Derivative superconvergence results: 1.2.1. ξ- or η- direction: tensor-product of triangular derivative superconvergence points in ξη plane and 1D Lobatto points in ζ dircection 1.2.2. ζ- direction: tensor-product of triangular function value superconvergence points in ξη plane and 1D Gaussian points in ζ dircection

32. Finite Element Superconvergence Pentahedral Elements 2. Serendipity family Vr where P(r+1)\Vr = Span{ξr+1-i-jηiζj| i,j,i+j = 0…r+1, j  1,r} • 2.1. Function value superconvergence results: • r = 2: the same as the tensor-product element • r = 3, 5: no superconvergence point • r = 4: mesh symmetry points

33. Finite Element Superconvergence Pentahedral Elements • 2.2. Derivative superconvergence results: • r = 1, 2: the same as the tensor-product elements • r = 3: • ξ-derivative: two lines • ζ-derivative: for Poisson equation, the plane • for Laplace equation, the plane and 4 points • r = 4: no superconvergence point • r = 5: • ξ-derivative: 6 points • ζ-derivative: 9 points

34. Finite Element Superconvergence Tetrahedral Elements Scheme 1 Scheme 2

35. Finite Element Superconvergence Tetrahedral Elements Master cell for scheme 2

36. Finite Element Superconvergence Tetrahedral Elements: Scheme 1 • Function value superconvergence results: • r = 2,4: vertices and midpoint of edges • r = 3: no superconvergence point • ξ-derivative superconvergence results: • r = 1,3: midpoint of edges parallel to the ξ-axis • (mesh symmetry points) • r = 2: 2nd order Gaussian points of edges parallel • to the ξ-axis • r = 4: no superconvergence point

37. Finite Element Superconvergence Tetrahedral Elements: Scheme 2 • Function value superconvergence results: • r = 2,4: vertices and midpoint of diagonal edges • r = 3: no superconvergence point • ξ-derivative superconvergence results: • No superconvergence point for Poisson equation. • For Laplace equation: • r = 1,3: midpoint of edges parallel to the ξ-axis • (non-mesh symmetry points) • r = 2: 2nd order Gaussian points of edges parallel • to the ξ-axis • r = 4: no superconvergence point

38. References Finite Element Superconvergence • Main References • I. Babuška and T. Strouboulis, The Finite Element Method and its Reliability. Oxford University Press, London, 2001. • I. Babuška, T. Strouboulis, C.S. Upadhyay, and S.K. Gangaraj, Computer based proof of the existence of Superconvergence points in the finite element method; Superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's and elasticity equations, Numer. Methods for PDEs, Vol.12, pp.347-392, 1996. • M. Krizek, P. Neittaanmaki, and R. Stenberg (Eds.) Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates. Lecture Notes in Pure and Applied Mathematics Series, Vol.196, Marcel Dekker, New York, 1997 • R. Lin and Z. Zhang, Natural superconvergence points of triangular finite elements. Numer. Meth. PDEs. 20 (2004), no. 6, pp. 864–906 • R. Lin and Z. Zhang, Derivative superconvergence of equilateral triangular finite elements. Accepted for publication by AMS Contemporary Mathematics Series • A.H. Schatz, I.H. Sloan, and L.B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. Vol. 33, No.2, pp. 505-521, 1996.

39. Finite Element Superconvergence References • Main References • B. Szabó, I. Babuška, Finite Element Analysis, John Wiley & Sons, New York, 1991. • L.B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, Vol. 1605, Springer, Berlin, 1995. • Z. Zhang, Derivative superconvergence points in finite element solutions of Poisson's equation for the serendipity and intermediate families -- A theoretical justification, Math. Comp., Vol. 67, pp. 541-552, 1998. • Z. Zhang, Derivative superconvergent points in finite element solutions of harmonic functions -- A theoretical justification, Math. Comp., Vol. 71, pp., 1421-1430, 2001. •   Z. Zhang and R. Lin, Locating natural superconvergent points of finite element methods in 3D. International Journal of Numerical Analysis and Modeling, 2 (2005), pp. 19-30 • R. Lin and Z. Zhang, Natural superconvergent points in 3D finite elements,SIAM Journal on Numerical Analysis 46-3 (2008), 1281-1297.