Sparse Tensor Galerkin Discretizations for First Order Transport Problems

Similar documents
Space-time sparse discretization of linear parabolic equations

Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs

Preconditioned space-time boundary element methods for the heat equation

Trefftz-discontinuous Galerkin methods for time-harmonic wave problems

Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances

Scientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1

From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes

WAVELETS, SHEARLETS AND GEOMETRIC FRAMES: PART II

LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)

Isogeometric Analysis:

Lecture 8: Boundary Integral Equations

From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes

Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.

Schwarz Preconditioner for the Stochastic Finite Element Method

The Helmholtz Equation

Scientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1

Deep Neural Networks and Partial Differential Equations: Approximation Theory and Structural Properties. Philipp Christian Petersen

Adaptive Petrov-Galerkin methods for first order transport equations. Author(s): Dahmen, Wolfgang; Huang, Chunyan; Schwab, Christoph; Welper, Gerrit

Computation of operators in wavelet coordinates

1 Discretizing BVP with Finite Element Methods.

Overlapping Schwarz preconditioners for Fekete spectral elements

A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations

Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Multiscale methods for time-harmonic acoustic and elastic wave propagation

Integral Representation Formula, Boundary Integral Operators and Calderón projection

Hierarchical Parallel Solution of Stochastic Systems

Robust exponential convergence of hp-fem for singularly perturbed systems of reaction-diffusion equations

Modelling in photonic crystal structures

Weight-adjusted DG methods for elastic wave propagation in arbitrary heterogeneous media

Fast and accurate methods for the discretization of singular integral operators given on surfaces

Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials

Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations

Finite Elements. Colin Cotter. February 22, Colin Cotter FEM

Fast Structured Spectral Methods

Algorithms for Scientific Computing

Adaptive algebraic multigrid methods in lattice computations

Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36

Tensor Sparsity and Near-Minimal Rank Approximation for High-Dimensional PDEs

Fast evaluation of boundary integral operators arising from an eddy current problem. MPI MIS Leipzig

Stabilization and Acceleration of Algebraic Multigrid Method

Newton-Multigrid Least-Squares FEM for S-V-P Formulation of the Navier-Stokes Equations

Multilevel stochastic collocations with dimensionality reduction

Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation

for compression of Boundary Integral Operators. Steven Paul Nixon B.Sc.

Weak Galerkin Finite Element Scheme and Its Applications

Two-scale Dirichlet-Neumann preconditioners for boundary refinements

Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations

Krylov subspace methods for linear systems with tensor product structure

Spline Element Method for Partial Differential Equations

Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions

Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms

Standard Finite Elements and Weighted Regularization

Maximum norm estimates for energy-corrected finite element method

Vector and scalar penalty-projection methods

Numerical Solution I

A Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems

Matrix assembly by low rank tensor approximation

A High-Order Galerkin Solver for the Poisson Problem on the Surface of the Cubed Sphere

Discontinuous Galerkin Methods

Computer simulation of multiscale problems

Chapter 12. Partial di erential equations Di erential operators in R n. The gradient and Jacobian. Divergence and rotation

On the interplay between discretization and preconditioning of Krylov subspace methods

Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques

1. Let a(x) > 0, and assume that u and u h are the solutions of the Dirichlet problem:

Final Ph.D. Progress Report. Integration of hp-adaptivity with a Two Grid Solver: Applications to Electromagnetics. David Pardo

A Finite Element Method for the Surface Stokes Problem

ADAPTIVE ANISOTROPIC PETROV-GALERKIN METHODS FOR FIRST ORDER TRANSPORT EQUATIONS

INTRODUCTION TO FINITE ELEMENT METHODS

Dual-Primal Isogeometric Tearing and Interconnecting Solvers for Continuous and Discontinuous Galerkin IgA Equations

Fast Sparse Spectral Methods for Higher Dimensional PDEs

Fast Algorithms for the Computation of Oscillatory Integrals

Projected Schur Complement Method for Solving Non-Symmetric Saddle-Point Systems (Arising from Fictitious Domain Approaches)

Solving PDEs with Multigrid Methods p.1

Hybridized Discontinuous Galerkin Methods

hp-dg timestepping for time-singular parabolic PDEs and VIs

1. Introduction. The Stokes problem seeks unknown functions u and p satisfying

Spherical Wave Approach (SWA) for the electromagnetic modelling of 3D GPR scenarios. Nicola Tedeschi

The Mortar Boundary Element Method

Iterative methods for positive definite linear systems with a complex shift

Adaptive Finite Element Methods Lecture Notes Winter Term 2017/18. R. Verfürth. Fakultät für Mathematik, Ruhr-Universität Bochum

New DPG techniques for designing numerical schemes

Numerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value

Trefftz-DG solution to the Helmholtz equation involving integral equations

Sparse Tensor Methods for PDEs with Stochastic Data II: Convergence Rates of gpcfem

arxiv: v2 [math.na] 8 Sep 2017

Efficient Augmented Lagrangian-type Preconditioning for the Oseen Problem using Grad-Div Stabilization

The All-floating BETI Method: Numerical Results

AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends

Isogeometric mortaring

Nonlinear, Transient Conduction Heat Transfer Using A Discontinuous Galerkin Hierarchical Finite Element Method

Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis

Contents as of 12/8/2017. Preface. 1. Overview...1

Spectral element agglomerate AMGe

OUTLINE ffl CFD: elliptic pde's! Ax = b ffl Basic iterative methods ffl Krylov subspace methods ffl Preconditioning techniques: Iterative methods ILU

Chapter 1: The Finite Element Method

Error analysis and fast solvers for high-frequency scattering problems

Institut de Recherche MAthématique de Rennes

High order Galerkin appoximations for parametric second order elliptic partial differential equations

A very short introduction to the Finite Element Method

Stochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows

Transcription:

Sparse Tensor Galerkin Discretizations for First Order Transport Problems Ch. Schwab R. Hiptmair, E. Fonn, K. Grella, G. Widmer ETH Zürich, Seminar for Applied Mathematics IMA WS Novel Discretization Methods Oct 31-Nov06 2010 Acknowledgement: Swiss Nat l Science Foundation and ERC W. Dahmen and G. Kutyniok Priority Research Programme 1324 German Research Foundation (DFG).

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 2 / 38 Outline 1 Problem 2 Variational Formulations 3 Galerkin discretization Without boundary conditions With boundary conditions 4 Numerical experiments 5 Tensor Product Wavelet Galerkin Approach (Widmer etal. JCP 2008) Inflow Boundary: Example Preconditioning Subspace Correction Preconditioner Experiment I 6 Adaptive Approximation Adaptivity: Rationale Numerical Experiment II Numerical Experiment III 7 Conclusion

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 3 / 38 The Radiative Transfer Problem Stationary monochromatic radiative transfer problem: (s x + κ(x)) u(x, s) = κ(x)i b (x), (x, s) D S d S u(x, s) = 0, x Γ (s) := D {s : s n(x) < 0}. x D R d D bounded, Lipschitz, s S d S 5-dimensional phase space (d D, d S ) = (3, 2) D R d D bounded: zero inflow BCs Model for general, parametric transport problems (UQ for Neutron Transport, v x ψ(x, v; t)...) Numerical approaches: MCM, discrete ordinates, P N approximation Complexity h d D D h d S S, slow convergence due to curse of dimensionality Sparse tensorization: Wavelets in S d S ( Widmer et al. (2008) ) Current: Sparse Adaptive Tensor Approximation with 1 spectral discretization in S d S ( this talk) 2 hierarchic, directional representation systems in D ( W. Dahmen s talk)

Variational formulation cg unstable = a) Stabilization or b) Petrov Galerkin (Talk of W. Dahmen) Stabilized LSQ formulation (Manteuffel, Ressel, Starke: SINUM(2000)): Find u V 0 such that with a(u, v) = l(v) v V 0 a(u, v) := (ɛ(s x u + κu), s x v + κv) L2 (D S), ɛ(x) = 1 κ(x) l(v) := (ɛκi b, s x v + κv) L2 (D S d S ) V := { u L 2 (D S d S ) : s x u L 2 (D S d S ) } V 0 := {u V : u = 0 on D, s n(x) < 0} a(, ) is continuous and coercive on V 0 V 0 Energy Norm u A := a(u, u) Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 4 / 38

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 5 / 38 Discretization Cartesian product domain D S d S span V with product basis functions from tensor product (TP) space H 1 (D) L 2 (S d S ) Inflow Boundary s-dependent: TP structure broken Hierarchical function spaces allow for incremental resolution refinement: On D: Piecewise polynomials (here p = 1) on dyadically refined mesh TD l V l D := S p,1 (D, T l D) On S d S : Spherical harmonics S (d S) n,m V l D = V l 1 D W l D up to order l H 1 (D) V l S := P d S l = span{s (d S) n,m : n = 0,..., l; m = 1,..., m n,ds } L 2 (S d S ) V l S = V l 1 S W l S Finite dimensional Full Tensor Product (FTP) space: V L = VD L VS N = 0 l D L 0 l S N W l D D W l S S H 1 (D) L 2 (S d S )

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 6 / 38 Structure of FTP space Illustration with VD L=3 = S (1,1) ([0, 1] [0, 1], TD 3 N=3 ) and VS = P 1 3 W l S S W 0 D W 0 S W 0 D W 1 S W 0 D W 2 S W 0 D W 3 S W 1 D W 0 S W 1 D W 1 S W 1 D W 2 S W 1 D W 3 S W 2 D W 0 S W 2 D W 1 S W 2 D W 2 S W 2 D W 3 S W 3 D W 0 S W 3 D W 1 S W 3 D W 2 S W 3 D W 3 S W l D D

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 7 / 38 Structure of Sparse Tensor Product (STP) space Illustration with VD L=3 = S (1,1) ([0, 1] [0, 1], TD 3 N=3 ) and VS = P 1 3 W l S S W 0 D W 0 S W 0 D W 1 S W 0 D W 2 S W 0 D W 3 S W 1 D W 0 S W 1 D W 1 S W 1 D W 2 S W 1 D W 3 S W 2 D W 0 S W 2 D W 1 S W 2 D W 2 S W 2 D W 3 S W 3 D W 0 S W 3 D W 1 S W 3 D W 2 S W 3 D W 3 S W l D D

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 8 / 38 Sparse Tensor Product (STP) Space FTP Finite Element FTP space: Finite Element STP space: V L = ˆV L := 0 l D L 0 l S N 0 f (l D,l S ) L with cutoff function f : [0, L] [0, N] R. Dimensionality: W l D D W l D D W l S S W l S S, dim(v L ) N D N S dim( ˆV L ) N D log N S + N S log N D for f (l D, l S ) = l D + l S L N

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 9 / 38 Approximation properties Error estimates for approximation u L of u H s+1,t (D S), s [0, p], t N 0 : Full tensor case u L V L : u u L H 1,0 (D S) max{n t/d S S, N s/d D } u (H 1,t H 1+s,0 )(D S) Sparse tensor case u L ˆV L with f (l D, l S ) = l D + L log 2 (l S +1) log 2 (N+1) : u u L H 1,0 (D S) log 2 ( N 1/d D ) max{n t/d S S, N s/d D } u H 1+s,t (D S) Convergence rate (almost) retained while number of dofs reduced. To equilibrate contributions: N h s/t

Product basis functions satisfying boundary conditions Zero inflow conditions: u(x, s) = 0 for x D, s n(x) < 0 Physical basis functions and corresponding outflow regions: 1 1 1 1 0 1 1 0 1 1 0 1 x 0 0 x x 2 1 0 0 x x 2 1 2 0 0 π π π 2 s 0 2 0 2 s s π 3π 2 π 3π 2 π x 1 0 3π 2 Partition of sphere into angular regions S q, mesh T S = {S q } n S q=1 : S 2 π 2 S 1 π 0 S 3 S 4 3π 2 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 10 / 38

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 11 / 38 Tensor product space with boundary conditions To maintain TP structure and strongly satisfy b.c.s add restricted angular functions to VS L: Introduce space P d S,T S N of angular functions restricted to elements of mesh T S, e. g. (d S = 1) Compressed spherical harmonics Legendre polynomials on S q. Split physical spaces into boundary and non-boundary subspaces: V l D = V l D V l D,0 Tensorize V 0 -conforming combinations only: V0 L := VD,0 L P d S N V D L P d S,T S N Sparse version: ˆV L 0 := V L D,0 ˆ P d S N V L D ˆ P d S,T S N

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 12 / 38 Structure of TP spaces with boundary conditions Active dofs for L = 3, N = 8 for FTP and STP space (here d D + d S = 2 + 1): Physical dofs 0 10 20 30 40 50 60 70 80 FTP l D S S 1 S 2 S 3 S 4 90 0 20 40 60 80 Angular dofs 0 1 2 3 Physical dofs 0 10 20 30 40 50 60 70 80 STP 90 0 20 40 60 80 Angular dofs 1853 dofs 275 dofs

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 13 / 38 Solution of discretized variational problem Benefits of TP structure: Separable quadrature for stiffness matrices over D and S System of equations is matrix equation: B n UC n = L Matrix-vector multiplication faster CG to solve system (currently no preconditioning). To evaluate solution: Look at Relative error of u in X = H 1,0, A, L 2 -norm on D S: err X = u u L X / u X n

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 14 / 38 Numerical experiments D = [0, 1] [0, 1], S = S 1, κ(x) = 1. Experiment 1 Solution prescribed: u(x, ϕ) = U(x)Y (ϕ) Experiment 2 U(x) = ( 1(x 1 1) 2 + 1)( 4(x 2 1/2) 2 + 1)x 1 x 2 ( 4(x 2 1)) 2 π ϕ + 1 if 0 ϕ < π 2 2 Y (ϕ) = π (ϕ 3π 2 ) if 3π 2 < ϕ < 2π 0 else U(x) as in Exp. 1 ( ) 2 sin(8ϕ) Y (ϕ) = 8ϕ if 0 ϕ < π 2 3π 2 < ϕ < 2π 0 else

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 15 / 38 Convergence rates Experiment 1 10 0 Error u (testside2), FT/ST, N=32, L refined 10 0 Error u (testside2), FT/ST, L=5, N refined u u N / u 10 1 10 2 10 3 H10 error FT A error FT L2 error FT H10 error ST A error ST L2 error ST 10 2 10 3 10 4 10 5 # Dofs p=2 p=1/2 p=1 u u N / u 10 1 10 2 10 3 In this setup: error limited by physical contribution p=2 10 3 10 4 10 5 # Dofs In physical space convergence rate 1/2 for H 1,0 -error as expected In angular space resolution refinement not effective ST method achieves better error per employed dofs ratio p=1/2 p=1

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 16 / 38 Convergence rates Experiment 2 10 0 Error u (testside1), FT/ST, N=32, L refined 10 0 Error u (testside1), FT/ST, L=5, N refined u u N / u 10 1 10 2 H10 error FT A error FT L2 error FT H10 error ST A error ST L2 error ST 10 2 10 3 10 4 10 5 # Dofs p=2 p=1/2 p=1 u u N / u 10 1 10 2 Here error limited by angular contribution Physical refinement not effective H10 error FT A error FT L2 error FT H10 error ST A error ST L2 error ST p=2 10 3 10 4 10 5 Angular convergence rate between 1/2 and 1 for H 1,0 -error # Dofs p=1/2 ST LSQ FEM achieves better error per dofs ratio for angular refinement p=1

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 17 / 38 Tensor Product Wavelet Galerkin D R 2 disc, dimd = 2, d S = 3 Meshes T 3 D, T 3 S 2 :

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 18 / 38 Multiscale Decompositions Nested meshes T l D nested spaces V l D := S0 1 (T l D ) TS l nested spaces VS l := S 1 0 (TS l) Complement ( detail, hierarchical surplus ) Spaces V l D = W l D V l 1 D V l S = W l S V l 1 S

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 19 / 38 Multiscale Decompositions In D R d : + In S d 1 : + + + + + +

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 20 / 38 Multiscale Decompositions Full Tensor Galerkin trial/test space V L = VD L VS L = WD i W j S 0 i,j L Sparse tensor product space: V L := WD i W j S dim V L = O(ND L log NS L + NS L log ND) L 0 i+j L

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 21 / 38 Sparse Tensor Product Space 00 01 02 V L := WD i W j S 0 i+j L 10 11 20

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 22 / 38 Graded ST Space Problem: Homogeneous Essential BC s in Γ (s) destroy TP structure Solution: 1 Enforce Dirichlet BC s weakly via Multipliers 2 Enforce Dirichlet BC s strongly: local isotropic mesh refinement along Γ (s), s S 1. Graded ST space

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 23 / 38 φ Inflow Boundary: Example 7 6 5 y 4 3 x 2 γ(x) φ D 1 0 1 10 5 0 Tangent in x to D y 5 10 4 3 2 1 0 1 2 3 4 x

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 24 / 38 Graded Sparse Tensor Product Space D S1 gamma Augment V L 0 := V 0 V L by tensor product nodal basis functions (in neighborhood of γ) multiscale tensor product is dictionary, frame (no Riesz basis anymore) dd Moderate increase of #d.o.f. by a logarithmic factor L. dim V 0 L = O(L 2 ( M L ) O(LML ) ) Approximation rate (proved for n = 2, d = 1, 2): C > 0 : L : û L V L 0 : u û L S C log h L h L u H 2,1 (D S 2 ).

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner transformation in standard basis

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner transformation in standard basis solution of decoupled, low dimensional transport problems (AMG)

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner transformation in standard basis solution of decoupled, low dimensional transport problems (AMG) Effort d.o.f. (Overhead of AMG ignored)

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 26 / 38 Preconditioning: Numerical Experiment Regular ST space Absorption coefficient: κ(x) = 1 Blackbody radiation: I b (x) = e 4 x 2

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 27 / 38 Numerical Experiment I Throughout: 2.5 D setting, D = {x R 2 x < 1}, n = d = 2 smooth solution u(x, s) due to large (constant) absorption 10 Emission f(r) 9 8 7 6 5 4 3 2 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 distance to origin r Blackbody intensity f ( x ) 1 0.5 0 0.5 1 Heat flux q(x)

Numerical Experiment I Efficacy of ST a priori DOF selection criterion: 1.18e+00 1 Level in Solid Angle 2 1.09e 02 3 1.04e 03 Size of multiscale contributions Ch. Schwab (SAM, ETH Zurich) 1.13e 01 l1l 2 l 3 Level in Physical Space l 1.00e 04 149120 largest multiscale contributions of solution in full tensor product space Sparse Tensor Approximation Novel Discretization Methods 28 / 38

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 29 / 38 Numerical Experiment I Numerical solutions: Incident Radiation G(x 1,0) 120 100 80 60 40 Level 0 Full Level 1 Full Level 1 Sparse Level 2 Full Level 2 Sparse Level 3 Full Level 3 Sparse 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1 Incident radiation Net emission (div q)

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 30 / 38 Numerical Experiment I Convergence (T l D, T l S by uniform regular refinement): Relative error of the incident radiation in the L 2 (D)-norm Relative error of the heat flux in the L 2 (D)-norm

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 31 / 38 Adaptivity: Rationale Propagation of singularities No elliptic lifting theorems for radiative transfer BVP!

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 31 / 38 Adaptivity: Rationale Propagation of singularities No elliptic lifting theorems for radiative transfer BVP! Example (dimd = 2, d S emitting region = 1): small compact

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 31 / 38 Adaptivity: Rationale Propagation of singularities No elliptic lifting theorems for radiative transfer BVP! Example (dimd = 2, d S emitting region = 1): small compact Note: more smoothness with (isotropic) scattering

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 32 / 38 Numerical Experiment II highly non-uniform absorption/source u non-smooth flux Hea

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 33 / 38 Numerical Experiment II (Limited) efficacy of a priori selection criterion: Size of multiscale contributions 149120 largest multiscale contributions of solution in full tensor product space

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 34 / 38 Numerical Experiment II Numerical solutions: Incident radiation Net emission

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 35 / 38 Numerical Experiment II Convergence (T l D, T l S by uniform regular refinement): Relative error of the incident radiation in the L 2 (D)-norm Relative error of the heat flux in the L 2 (D)-norm

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 36 / 38 Numerical Experiment III { 5, x 0.08 Absorption: κ(x, y) = 3, otherwise. { e 20 x x 0 2 e Source: I b (x) = 5, x x 0 0.5 0, otherwise. x 0 = ( ) 0.08 0.02 Threshold ν = 0.04

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 37 / 38 Conclusion STP combined with spectral and MR discretization of S d S essential b.c.s on Γ (s) strongly satisfied & TP structure maintained STP method results in discretization with far fewer dofs than FTP Theoretical error estimates without b.c.s shown to be (almost) identical for FTP and STP Error estimates with b.c.s numerically confirm theoretical estimates Adaptive mesh refinements in D and in S d S Essential BC s and Directional Adaptivity: stable frame discretizations of transport problems in bounded domains D

Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 38 / 38 References Konstantin Grella, Christoph Schwab. Sparse tensor spherical harmonics approximation in radiative transfer. Technical report, SAM, ETH Zürich, 2010. www.sam.math.ethz.ch/ reports G. Widmer, R. Hiptmair, and Ch. Schwab. Sparse adaptive finite elements for radiative transfer. Journal of Computational Physics, 227:6071 6105, 2008. Gisela Widmer. Sparse Finite Elements for Radiative Transfer. PhD thesis, ETH Zürich, 2009. No. 18420.

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback