On the accuracy of dG discretizations on curved and agglomerated - PowerPoint PPT Presentation

Workshop on Polygonal and Polyhedral Meshes, Università di Milano Bicocca, 17-19 settembre 2012. On the accuracy of dG discretizations on curved and agglomerated elements meshes 1 1 1 Lorenzo Botti , Francesco Bassi , Alessandro Colombo . 1 Università degli Studi di Bergamo, Dipartimento di Ingegneria Industriale.

Outline 1. Compare physical and reference frame dG discretizations on curved elements meshes • L2-orthogonal projection (approximation properties). • INS dG discretization, Kovasznay flow (efficiency). 2. Evaluate the accuracy on agglomerated elements obtained clustering together the cells of a standard fine grid 3. Agglomeration based hemodynamic application Exploit the flexibility of agglomeration based dG discretizations in complex 3D domains

Motivation Why dG methods on general meshes? dG methods are well suited in the context of conservation laws • Conservation of physical quantities • Robustness in convection dominated flows • Accuracy near boundaries to resolve boundary layers Curved elements meshes are commonly employed to • Increase the accuracy of the domain discretization • Decrease the number of elements to perform higher-order discretizations Challenges we need to face... dG discretizations are expensive To justify the computational effort the accuracy should be maintained also on curved element meshes. High-order methods need accurate domain approximations Exploit the geometrical flexibility provided by agglomeration to avoid high-order mesh generation techniques.

Accuracy on general meshes Consider a mesh composed of quadrilateral elements . Discrete polynomial space choice • Physical frame discretizations (PFD) • Reference frame discretizations (RFD) where is a polynomial mapping such that and is the reference square. The discrete space choice might have a significant impact on the accuracy. h -convergence rates associated to reference frame discretization might degrade depending on • the element shape • the mapping function • the discrete polynomial space see Arnold, Boffi, Falk [Math. Comp., 2002].

Accuracy on general meshes, physical frame Physical frame discretizations provide optimal approximation properties on regular mesh sequences, that is where is the -orthogonal projection onto and the constant depends only on a mesh regularity parameter and the polynomial degree k. Optimal approximation properties for regular mesh sequences can be proved using the version of the Bramble-Hilbert Lemma introduced by Brenner and Scott [The mathematical theory of finite element methods, Springer 2008] , based on averaged Taylor polynomials. See also [Botti, J. Sci. Comput., 2012]. Mesh regularity implies that each mesh element E is star-shaped with respect to a ball with uniformly comparable diameter with respect to .

Accuracy on general meshes, physical frame Physical frame discretizations provide optimal approximation properties on regular mesh sequences, that is where is the -orthogonal projection onto and the constant depends only on a mesh regularity parameter and the polynomial degree k. Mesh regularity implies that each mesh element E is star-shaped with respect to a ball with uniformly comparable diameter with respect to .

Accuracy on general meshes, reference frame We define the effective mapping order associated to the polynomial space as the minimum positive integer such that , where . Examples curved 8-node quadrilateral, curved 9-node quadrilateral, mapping space mapping space

Accuracy on general meshes, reference frame We define the effective mapping order associated to the polynomial space as the minimum positive integer such that , where . Examples curved 9-node quadrilateral, 8-node rectangular element, mapping space effective mapping space

Accuracy on general meshes, reference frame We define the effective mapping order associated to the polynomial space as the minimum positive integer such that , where . The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection operator onto the space , where .

Accuracy on general meshes, reference frame We define the effective mapping order associated to the polynomial space as the minimum positive integer such that , where . The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection operator onto the space , where . Sketch of the proof We consider the approximation of a function v over an element E using the space , the error in norm can be computed as

Accuracy on general meshes, reference frame We define the effective mapping order associated to the polynomial space as the minimum positive integer such that , where . The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection operator onto the space , where . Since, as suggested by Arnold, Boffi, Falk [Math. Comp., 2002] we are able to infer that

Accuracy on general meshes, reference frame We define the effective mapping order associated to the polynomial space as the minimum positive integer such that , where . The approximation properties associated to the discrete polynomial space can be evaluated considering an -orthogonal projection operator onto the space , where . Using the effective mapping order definition we get see Botti [J. Sci. Comput., 2012] for details.

Accuracy on general meshes Compare the accuracy of physical and reference frame discretizations. Consider the L2 projection of y^6 on regular mesh sequences Distorted second-order quadrilateral Second-order quadrilateral meshes meshes of the unit square approximating the unit circle

Accuracy on general meshes, square mesh sequence Physical frame discretizations Reference frame discretizations k=0 1 k=0 nine-node quadrilaterals eight and nine-node quadrilaterals 0.1 k=1 k=1 0.1 0.01 k=2 k=2 0.01 k=3 0.001 k=3 0.001 k=4 error in L2 norm error in L2 norm 1 x 10 -4 k=4 1 x 10 -4 k=5 k=5 1 x 10 -5 k=6 1 x 10 -5 1 x 10 -6 k=7 1 x 10 -6 k=8 1 x 10 -7 1 x 10 -7 k=9 1 x 10 -8 1 x 10 -8 k=10 1 x 10 -9 1 x 10 -9 k=11 1 x 10 -10 k=12 1 x 10 -10 k=13 1 x 10 -11 1 x 10 -11 k=14 1 x 10 -12 1 x 10 -12 k=15 1 x 10 -13 10 100 1000 10 100 1000 sqrt(number of elements) sqrt(number of elements) k=0 eight-node quadrilaterals 0.1 k=1 0.01 k=2 0.001 k=3 error in L2 norm Convergence rates (error in L2 norm) 1 x 10 -4 k=4 k=5 1 x 10 -5 k=6 1 x 10 -6 k=7 • Reference frame discretizations 1 x 10 -7 k=8 1 x 10 -8 k=9 1 x 10 -9 k=10 • Physical frame discretizations 1 x 10 -10 k=11 1 x 10 -11 1 x 10 -12 1 x 10 -13 10 100 1000 sqrt(number of elements)

Accuracy on general meshes, circular mesh sequence. Reference frame discretizations Physical frame discretizations k=0 k=0 0.1 0.1 k=1 k=1 0.01 0.01 k=2 k=2 0.001 0.001 k=3 k=3 1 x 10 -4 1 x 10 -4 error in L2 norm error in L2 norm k=4 k=4 1 x 10 -5 1 x 10 -5 k=5 k=5 1 x 10 -6 1 x 10 -6 k=6 1 x 10 -7 1 x 10 -7 k=7 1 x 10 -8 1 x 10 -8 k=8 1 x 10 -9 1 x 10 -9 1 x 10 -10 1 x 10 -10 1 x 10 -11 1 x 10 -11 1 x 10 -12 1 x 10 -12 1 x 10 -13 1 x 10 -13 1 x 10 -14 1 x 10 -14 10 100 10 100 sqrt(number of elements) sqrt(number of elements) Since for most of the mesh elements , the convergence rates of reference frame discretizations are not spoiled.

Accuracy on general meshes, Circular mesh family. Reference frame discretizations Physical frame discretizations k=0 k=6 0.1 0.1 k=1 k=0 0.01 0.01 k=2 k=1 0.001 0.001 k=3 k=2 1 x 10 -4 1 x 10 -4 error in L2 norm error in L2 norm k=4 k=3 1 x 10 -5 1 x 10 -5 k=5 k=4 1 x 10 -6 1 x 10 -6 k=6 k=5 1 x 10 -7 1 x 10 -7 k=7 1 x 10 -8 1 x 10 -8 k=8 1 x 10 -9 1 x 10 -9 1 x 10 -10 1 x 10 -10 1 x 10 -11 1 x 10 -11 1 x 10 -12 1 x 10 -12 1 x 10 -13 1 x 10 -13 1 x 10 -14 1 x 10 -14 10 100 10 100 sqrt(number of elements) sqrt(number of elements) Since for most of the mesh elements , the convergence rates of reference frame discretizations are not spoiled. Physical frame discretizations provide the best accuracy for a given number of dofs.

Accuracy on general meshes, INS Compare the accuracy obtained on the Kovasznay flow problem with physical and reference frame dG discretizations. INS dG discretization proposed by Di Pietro and Ern, [Math. Comp., 2010]. Randomly distorted mesh sequence Element subdivision mesh sequence