Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems
|
|
- Edwina Ward
- 5 years ago
- Views:
Transcription
1 Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems Donald Estep University Interdisciplinary Research Scholar Department of Statistics Center for Interdisciplinary Mathematics and Statistics Colorado State University Research supported by Defense Threat Reduction Agency (HDTRA ), Department of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909, DE-SC , DE-SC , INL , DE SC9279), Idaho National Laboratory ( , ), Lawrence Livermore National Laboratory (B573139, B584647, B590495), National Science Foundation (DMS , DMS , DGE , MSPA-CSE , ECCS , DMS , DMS , DMS-FRG , DMS ), National Institutes of Health (#R01GM096192), Sandia Corporation (PO299784) March 26, 2014 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 1/77
2 Collaborators Todd Arbogast, UT Austin John Shadid, Sandia National Laboratory Simon Tavener, Colorado State University From my group: Varis Carey, UT Austin Jeb Collins, Colorado State University Victor Ginting, U. Wyoming Jehanzeb Hameed, Colorado State University Brendan Sheehan, Colorado State University Tim Wildey, Sandia National Laboratory Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 2/77
3 Multiphysics, Multiscale Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 3/77
4 Multiphysics, Multiscale Systems Multiphysics, multiscale systems couple different physical processes interacting across a wide range of scales Such systems abound in science and engineering application domains Computational modeling plays a critical role in the predictive study of such systems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 4/77
5 Example 1: A MEMs Thermal Actuator V sig Contact Contact Contact Conductor 250 mm Conductor V switch (σ V ) = 0 (κ(t ) T ) = σ( V V ) ˆ (λ tr(e)i + 2µE β(t T ref )I ) = 0 E = ( ˆ d + ˆ d ) /2 electrostatic current energy displacement This is an example of parameter passing, in which the solution of one component is used to compute the parameters and/or data for another component Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 5/77
6 Example 2: Reaction-Diffusion Equations The Brusselator problem u 1 t ɛ u 1 = α βu 1 + u 2 1 u 2, u 2 t ɛ u 2 = γu 1 u 2 1 u 2, suitable initial and boundary conditions The Brusselator is a simple model of the reaction kinetics of the two main species in an autocatalytic reaction taking place in solution Tight coupling of different physical components with different scales and stability properties, e.g. reaction and diffusion, in the same equations Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 6/77
7 Example 3: Fluid-Solid Conjugate Heat Transfer Solid µ u + ρ 0 (u ) u + p + ρ 0 βt F g = ρ 0 (1 + βt 0 ) g, u = 0, k F T F + { ρ 0 c p (u T F ) = Q F, T S = T F, interface k F (n T F ) = k S (n T S ), k S T S = Q S, Physics in different regions are coupled through a common interface Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 7/77
8 Example 4: Physics in a Fusion Reactor Simplified model for core-edge coupling T c t = x χ c Tc x + S c, core region T m t = x χ m Tm x + S m ν md (T m T d ), edge midplane region T d t = x χ d T d x + S d + ν md (T m T d ) ν el T d θ(x x s ), edge divertor region The edge and core regions are coupled through a common boundary: { T c = 1 2 (T m + T d ), χ c T c n = 1 2 (χ m T m n + χ d T d n) with S c = S m + S d and χ c = χ m + χ d at the interface The physics is coupled across scales in space and time Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 8/77
9 Multiscale Operator Decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 9/77
10 Multiscale Operator Decomposition Multiscale, multiphysics systems pose severe challenges for computational solution: Scale effects Complex stability Coupling between physics Different kinds of physical descriptions Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 10/77
11 Multiscale Operator Decomposition Multiscale operator decomposition (MOD) methods are widely used for solving multiphysics, multiscale systems The system is decomposed into components representing relatively simple physics and/or single scales and a global solution is sought by iterating solutions of the components Fully Coupled Multiscale Operator Decomposition Solution Physics 1 Physics 1 1 Physics 1 2 Physics 1 3 upscale downscale upscale downscale upscale downscale upscale downscale Physics 2 Physics 2 1 Physics 2 2 Physics 2 3 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 11/77
12 MOD for a Coupled Elliptic System A simplified Thermal Actuator model: a 1 u 1 + b 1 u 1 + c 1 u 1 = f 1 (x), x Ω, a 2 u 2 + b 2 u 2 + c 2 u 2 = f 2 (x, u 1, Du 1 ), x Ω, u 1 = u 2 = 0, x Ω, Ω is a bounded domain with boundary Ω The coefficients are smooth functions and a 1, a 2 are bounded away from zero We wish to compute information depending on u 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 12/77
13 MOD for a Coupled Elliptic System Algorithm Construct discretizations T h,1, T h,2 and finite element spaces S h,1, S h,2 Compute a finite element solution U 1 S h,1 (Ω) of the first equation Project U 1 S h,1 (Ω) into the space S h,2 (Ω) Compute a finite element solution U 2 S h,2 (Ω) of the second equation The projection between the discretization spaces is a crucial step In a fully coupled system, U 2 would be projected into the space for U 1 for the next iteration Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 13/77
14 MOD for a Reaction-Diffusion Equation We consider the reaction-diffusion problem { du dt = u + F (u), 0 < t, u(0) = u 0 The diffusion component u induces stability and change over long time scales The reaction component F induces instability and change over short time scales Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 14/77
15 MOD for a Reaction-Diffusion Equation Multiscale Operator-Splitting Algorithm On (t n 1, t n ], use the fine mesh to solve { du R dt = F (u R ), t n 1 < t t n, u R (t n 1 ) = u D (t n 1 ) On (t n 1, t n ], use the coarse mesh to solve { du D dt (u D ) = 0, t n 1 < t t n, u D (t n 1 ) = u R (t n ) Advance to the next diffusion step The operator split approximation is u(t n ) u D (t n ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 15/77
16 MOD for a Reaction-Diffusion Equation To account for the fast reaction, we use a multirate approach We solve the reaction component using many time steps inside each diffusion step Diffusion Integration: Reaction Integration: t t t t t t 0 t 1 t t 3 t 4 t 5 s1... s 1,0 s 1,M... s 2,0 s 2,M s 2 s 3... s 3,0 s 3,M... s 4,0 s 4,M Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 16/77
17 MOD for Coupling Through a Boundary A parabolic interface problem { u1 t (χ 1 u 1 ) = f 1, x Ω 1 u 2 t (χ 2 u 2 ) = f 2, x Ω 2 with coupling conditions { u 1 = u 2, χ 1 u 1 n = χ 2 u 2 n, x Γ We use different space and time discretizations for the two components We introduce projections between the discrete spaces on the two sides of the interface Γ Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 17/77
18 MOD for Coupling Through a Boundary Let T n i denote an approximation at time t n. Gauss-Seidel Algorithm Given u n 1 2, solve for u n 1 with un 1 = un 1 2 on the interface Given u n 1, solve for un 2 with χ 2 u n 2 n = χ 1 u n 1 n on the interface Move on to t n+1 There are many other ways to arrange iterations between components and timesteps Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 18/77
19 The Effects of Error from Multiscale Operator Decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 19/77
20 Issues with Multiscale Operator Decomposition Multiscale operator decomposition discretizes the instantaneous interactions between the component physics The price: Transfer of computed information between components affects the accuracy of the numerical solution New forms of instability, some of which are very subtle, arise Even if each physical component is resolved accurately, decomposition may cause loss of fidelity These consequences are often not apparent on either the component or global level. Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 20/77
21 A Simple Thermal Actuator Consider u 1 = sin(4πx) sin(πy), x Ω u 2 = b u 1 = 0, x Ω, u 1 = u 2 = 0, x Ω, b = 2 π ( ) sin(4πx) 25 sin(πy) where Ω = [0, 1] [0, 1] We consider the quantity of interest u 2 (.25,.25) We solve for u 1 first and then solve for u 2 using independent meshes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 21/77
22 A Simple Thermal Actuator Using uniform meshes, an a posteriori error estimate yields estimate of the error in the quantity of interest.0042 true error.0048 discrepancy in estimate.0006 ( 13%) u u This arises from the operator decomposition Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 22/77
23 A Simple Thermal Actuator Adapting the mesh using only an error estimate for the second component causes the discrepancy to become alarmingly worse estimate of the error in the quantity of interest.0001 true error u Pointwise Error of u Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 23/77
24 The Brusselator Problem u 1 t u 1 = f 1 (u 1, u 2 ) = 0.6 2u 1 + u 2 1 u 2, u 2 t u 2 = f 2 (u 1, u 2 ) = 2u 1 u 2 1 u 2, suitable initial and boundary conditions Linear finite element method in space with 500 elements A standard first order splitting scheme Trapezoidal Rule with time step of.2 for the diffusion and Backward Euler with time step of.004 for the reaction Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 24/77
25 The Brusselator Problem 10 0 L 2 norm of error slope t t = 6.4 t = 16 t = 32 t = 64 t = 80 Operator Split Solution Spatial Location On moderate to long time intervals, there is a critical time step above which convergence fails Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 25/77
26 Model for Core-Edge Coupling Through a Boundary A parabolic interface problem { u1 t (χ 1 u 1 ) = f 1, x Ω 1 u 2 t (χ 2 u 2 ) = f 2, x Ω 2 with coupling conditions { u 1 = u 2, χ 1 u 1 n = χ 2 u 2 n, x Γ Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 26/77
27 Model for Core-Edge Coupling Through a Boundary 0.4 Iteration and Transfer Error Time Using t = 10 3, the total error of the approximation increases with time Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 27/77
28 Model for Core-Edge Coupling Through a Boundary Iteration Error Time Transfer Error Time The error arising from incomplete iteration on each step becomes negligible as time passes The transfer error accumulates with time and becomes the largest source of error Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 28/77
29 Model for Core-Edge Coupling Through a Boundary log( L 2 error) Fully Coupled Solution Operator Decomposition log(sqrt(degrees of freedom)) Though we use second order accurate methods for each component, the error in the operator decomposition approximation is only first order in space Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 29/77
30 A Posteriori Error Estimation using Residuals and Adjoint Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 30/77
31 A Posteriori Error Analysis We estimate the error in a quantity of interest computed from numerical solutions Ingredients: We compute residuals to quantify the introduction of error, e.g. through discretization We solve the adjoint problem associated with the quantity of interest to quantify the effects of stability We use variational analysis to combine residuals and adjoint solutions to obtain an estimate The goal is different than convergence analysis: We seek accurate estimates of error, not general but inaccurate bounds Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 31/77
32 A Linear Algebra Example Let U be an approximate solution of Au = b We compute a quantity of interest given by a linear functional (ψ, u) The error is not computable e = u U We can compute the residual R, R = AU b Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 32/77
33 A Linear Algebra Example The adjoint problem for the generalized Green s vector is A φ = ψ Variational analysis: (ψ, e) = (A φ, e) = (φ, Ae) = (φ, R) This gives the error representation: (ψ, e) = (φ, R) We solve for φ numerically to compute the estimate Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 33/77
34 Condition Numbers and Adjoint Solutions The classic error bound is e Cκ(A) R The condition number κ(a) = A A 1 is a measure of stability The a posteriori estimate yields (e, ψ) φ R The stability factor φ is a weak condition number for the quantity of interest We can obtain κ from φ by taking the sup over all ψ = 1 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 34/77
35 Condition Numbers and Adjoint Solutions Compute (y, e 1 ) from the solution of Ay = b where A is a random matrix Condition number of A is estimate of the error in the quantity of interest a posteriori error bound for the quantity of interest traditional error bound for the error Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 35/77
36 Abstract A Posteriori Error Analysis We solve a differential equation L(u) = f for a quantity of interest given by a linear functional (u, ψ) ψ determines the quantity of interest (, ) denotes the L 2 inner product over space and time We compute a numerical approximation U u The local resolution of the approximation is measured using the residual (written formally as) R(U) = L(U) f Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 36/77
37 Abstract A Posteriori Error Analysis To build an error estimate, we solve a linear adjoint problem (DL) φ = ψ We obtain the error representation (U u, ψ) = (R(U), W(φ)) for a suitable adjoint weight W(φ) The adjoint weight reflects stability by scaling residuals to create local contributions to the error The estimate accounts for the accumulation, cancellation, and propagation of local error contributions to the global error The stability information is specific to the quantity of interest Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 37/77
38 A Posteriori Analysis for an Elliptic Problem Let U be a Galerkin finite element approximation for a u + b u + cu = f on Ω The residual in weak form is (R, v) = (a U, v)+ ( b U+cU f, v ) for all test functions v Obtained by integration by parts, the formal adjoint problem is a φ div(bφ) + cφ = ψ on Ω The estimate is (e, ψ) = ( a U, (I π h )φ) ) + ( b U + cu f, (I π h )φ ) π h is a projection into the space of U Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 38/77
39 An Estimate for an Elliptic Problem { u = 200π 2 sin(10πx) sin(10πy), (x, y) Ω = [0, 1] [0, 1], u = 0, The solution is u = sin(10πx) sin(10πy) (x, y) Ω 1 u 0-1 x y error/estimate percent error Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 39/77
40 Using Adjoints to Quantify Stability The adjoint problem quantifies the effects of stability These effects are specific to the quantity of interest ((.05 + tanh(10(x 5) (y 1) 2 )) u ) ( ) u = 1, (x, y) Ω = [0, 10] [0, 2], 0 u = 0, (x, y) Ω Convection Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 40/77
41 Using Adjoints to Quantify Stability 6 1 x y x x y x 8 10 Final meshes for an average error of 4% 24,000 elements versus 3500 elements Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 41/77
42 Analysis of Multiscale Operator Decomposition for Multiscale, Multiphysics Problems Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 42/77
43 Analysis of Multiscale Operator Decomposition We extend the a posteriori analysis to deal with three issues arising in MOD: The passing of error along with information between components of a multiphysics model Defining appropriate adjoint operators becomes complicated Errors are introduced in complicated ways, which complicates how error can be mitigated Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 43/77
44 Multiscale Operator Decomposition: Issue 1 Computed Information Physics 1 Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 44/77
45 Multiscale Operator Decomposition: Issue 1 Computed Information Error Physics 1 Physics 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 45/77
46 Multiscale Operator Decomposition: Issue 1 Computed Information Error Physics 1 Physics 2 We define auxiliary quantities of interest corresponding to information passed between components We solve auxiliary adjoint problems to estimate the error in that information In an iterative scheme, we also estimate the history of errors passed from one iteration level to the next We can also estimate the effect of processing, e.g. up and down scaling, the information Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 46/77
47 A Linear Algebra Example Consider the triangular system ( ) ( ) A11 0 u1 Au = = b = A 21 A 22 u 2 with approximate solution U = ( U1 U 2 ) ( u1 u 2 ( b1 b 2 ) ) = u We estimate the error in a quantity of interest in u 2 given by ( ψ (1), u ) = ( ψ (1) 2, u ) 2 Estimates on u 1 are independent of u 2 We use a superscript because further estimates are required Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 47/77
48 A Linear Algebra Example The lower triangular structure of A yields A 11 u 1 = b 1, A 22 u 2 = b 2 A 21 u 1 The residuals are R 1 = b 1 A 11 U 1, R 2 = (b 2 A 21 u 1 ) A 22 U 2 The residual of U 2 depends on information from the first component Decreasing the residual of U 2 requires consideration of the accuracy of the first component Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 48/77
49 A Linear Algebra Example The adjoint problem reads ( A 11 A 12 0 A 22 The error representation is ) ( ) φ (1) 1 φ (1) = 2 ( ) 0 ψ (1) 2 ( ψ (1), e ) = ( φ (1) 2, R ) ( (1) 2 + φ 2, A ) 21e 1 This only uses φ (1) 2 and stays in the single physics paradigm The first term on the right is computable as usual The second term on the right represents the effect of errors in the first component on the second component Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 49/77
50 A Linear Algebra Example We have to compute the decomposition term ( (1) φ 2, A ) ( 21e 1 = A 21 φ (1) 2, e ) 1 The first adjoint problem does not estimate this error This is a linear functional of the error e 1 corresponding to adjoint data ( ) ( ) ψ (2) ψ (2) = 1 A = 21 φ(1) We pose a secondary adjoint problem ( A 11 A ) ( ) ( ) 12 φ (2) 0 A 1 ψ (2) 22 φ (2) = Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 50/77
51 A Linear Algebra Example The block structure of A yields φ (2) 2 = 0, A 11φ (2) 1 = ψ (2) 1 = A 21φ (1) 2 Note these solves stay in the single physics paradigm We obtain the secondary error representation ( ψ (2), e ) = ( φ (2) 1, b ) ( (2) 1 A 11 U 1 = φ 1, R ) 1 The complete error representation is ( ψ (1), e ) = ( φ (1) 2, R ) ( (2) 2 + φ 1, R ) 1 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 51/77
52 A Linear Algebra Example Compute the average of the coefficients of (u 1, u 2 ) solving: ( ) ( ) A11 0 u1 = A 21 A 22 u 2 ( b1 b 2 ), A 11, A 21, A 22, b 1, b 2 U( 1, 1) A 11 = I +.2A 11 A 21 = A 21, A 22 = I +.1A 22 We use blockwise Gauss-Seidel iteration ( ψ (1), e ) iterations for U 1 40 or more iterations for U 2 ( (1) φ 2, R ) ( (2) φ 1, R ) The error in the QoI is mainly due to transferred error If we double the iterations for U 1, the error in the QoI is 10 7 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 52/77
53 Analysis for an Elliptic System Recall the triangular problem u 1 = sin(4πx) sin(πy), x Ω u 2 = b u 1 = 0, x Ω, u 1 = u 2 = 0, x Ω, b = 2 π ( ) 25 sin(4πx) sin(πx) where Ω = [0, 1] [0, 1] We consider the quantity of interest u 2 (.25,.25) We solve for u 1 first and then solve for u 2 using independent meshes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 53/77
54 A Simplified Model Problem u 1 = f 1 (x), x Ω, u 2 = f 2 (x, u 1, Du 1 ), x Ω u 1 = 0, u 2 = 0, x Ω We compute a quantity of interest ( ψ (1), u ) that only depends on u 2, thus ( ) 0 ψ (1) = ψ (1) 2 We require the linearization Lf 2 (w) of f 2 with respect to u 1 around the function w Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 54/77
55 The First Adjoint Problem The weak form of the primary adjoint problem is {( (1) φ 1, v ) ( 1 + Lf2 (U 1 )φ (1) 2, v 1) = 0, ( (1) φ 2, v ) ( (1) 2 = ψ 2, v 2), all test functions v 1, v 2 This yields the first error representation ( ψ (1), e ) = ( f 2 (u 1, Du 1 ), (I π h )φ (1) 2 ) ( U2, (I π h )φ (1) ) 2 The residual depends on the unknown true solution We write f 2 (u 1, Du 1 ) = f 2 (U 2, DU 2 ) + ( f 2 (u 1, Du 1 ) f 2 (U 2, DU 2 ) ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 55/77
56 The Secondary Adjoint Problems We have a new linear functional of the error ( f2 (u 1, Du 1 ) f 2 (U 1, DU 1 ), φ (1) ) ( 2 Lf(U1 )e 1, φ (1) ) 2 = ( Lf(U 1 ) φ (1) 2, e ) ( 1 = ψ (2), e ) We pose a secondary adjoint problem {( (2) φ 1, v ) ( 1 + Lf2 (U 1 )φ (2) 2, v ) ( 1 = Lf2 (U 1 ) φ (1) 2, v ) 1 ( (2) φ 2, v 2) = 0 all test functions v 1, v 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 56/77
57 Representation Formulas We finally get the fully computable representation Theorem ( ψ (1), e ) ( f 2 (U 1, DU 1 ), (I π h )φ (1) ) ( U2, (I π h )φ (1) ) + ( f 1, (I π h )φ (2) 1 2 ) ( U1, (I π h )φ (2) 1 φ (1), φ (2) are the primary and secondary adjoint solutions The new analysis estimates the error in the information passed between components 2 ) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 57/77
58 An Elliptic System If we adapt the meshes using all the terms in the estimate, we can drive the error below We actually refine the mesh for u 1 more than the mesh for u 2 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 58/77
59 Further Extensions of the Representation Formula The analysis can be extended further: If we use different meshes and/or numerical methods in the different components, then there is error arising from projection between the representations We define an tertiary auxiliary quantity of interest and related adjoint problem and add another term in the estimate In a fully coupled system, there are additional terms estimating the errors passed between iterations and the error arising from incomplete iteration Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 59/77
60 Multiscale Operator Decomposition: Issue 2 The adjoint operator associated to a multiscale operator decomposition solution is generally different than the adjoint operator associated with the full problem There are difficult technical issues about defining the appropriate adjoint operators The standard approach to define an adjoint for a nonlinear problem fails Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 60/77
61 The Standard Approach for Nonlinear Problems Consider U u solving With residual R(U) = F (U) b, F (u) = b F (u) F (U) = R(U) This gives the linearized representation A e = R(U), A = F (u, U) We carry out the a posteriori analysis using the adjoint associated with A Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 61/77
62 The Standard Approach Fails for MOD A MOD approximation Ũ solves a different problem Ũ approximately solves an operator decomposed problem F (ũ) = b, R( Ũ) = F (Ũ) b Now we obtain F (u) F (Ũ) }{{} cannot linearize = R(Ũ) }{{} can be made small Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 62/77
63 The Standard Approach Fails for MOD The solution is problem-specific But in every case, this introduces an error in the adjoint operator associated with the MOD solution The estimates for MOD include terms that quantify the error in the adjoint operator This measures the impact of operator decomposition on stability Additional work is needed to obtain a computable estimate Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 63/77
64 Operator Splitting for Reaction-Diffusion Equations To treat this case, we assume that the full operator and the operator decomposed operator share a common solution, e.g. a steady-state We linearize the full and operator decomposed operator individually with respect to the common solution We carry out the analysis by comparing the resulting linear equations Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 64/77
65 Operator Splitting for Reaction-Diffusion Equations We derive a new type of hybrid a priori - a posteriori estimate Theorem Error in the Quantity of Interest Q 1 + Q 2 + Q 3 Q 1 estimates the error of the numerical solution of each component in the standard way Q 2 N n=1 (U n 1, E n 1 ), E a computable estimate for the error in the adjoint arising from operator decomposition Q 3 = O( t 2 ) is an a priori expression that is provably higher order Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 65/77
66 The Brusselator Problem Accuracy of the error estimate t = 0.01, M = 10 t = 0.01, M = 10 Error Error Component Component T = 8 T = 40 Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 66/77
67 Multiscale Operator Decomposition: Issue 3 Correcting the effects of decomposition efficiently requires some thought Simply using mesh refinement may not be an efficient approach Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 67/77
68 Conjugate Heat Transfer for a Fluid-Solid System Theorem (ψ, e)= (f, φ π V φ) a 1 (u {k} h, φ π V φ 1 ) c 1 (u {k} h, u{k} h, φ π V φ) b(φ π V φ, p h ) d(t {k} F,h, φ π V φ) b(u {k} h, z π Zz) + ( Q F, θ F π WF θ F ) a2 (T {k} F,h, θ F π WF θ F ) c 2 (u {k} h, T {k} F,h, θ F π WF θ F ) + ( Q S, θ S π WS θ S ) a3 (T {k} S,h, θ S π WS θ S ) + ( T {k} S,h π ST {k} F,h, k S(n θ S ) ) + ( π Γ S T {k} I F,h T {k} F,h, k S(n θ S ) ) Γ I + ( k S (n T {k} S,h ), π ) W F θ F + ( ) ( {k} Q Γ S, π WS θ S a3 T I S,h, π ) W S θ S The estimate involves auxiliary quantities of interest and adjoint problems measuring errors passed between components and between iteration levels Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 68/77
69 Conjugate Heat Transfer One specific term in the estimate causes the observed loss of order of accuracy It arises from passing numerical fluxes at the interface We can mitigate this several ways Use higher order elements or highly refined meshes Compute a discrete boundary flux correction for the error in the derivative on the interface (Wheeler, Carey) The last post-processing approach is very cheap and can be applied easily to existing codes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 69/77
70 Conjugate Heat Transfer Between a Fluid and a Solid Quantity of interest is the temperature at a small region in the center of the cylinder No flux correction is applied Conditions are set to simulate the flow of water past a cylinder made from stainless steel We use Taylor-Hood elements to solve the fluid and standard finite element methods for the solid with independent meshes Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 70/77
71 Conjugate Heat Transfer Between a Fluid and a Solid Using the inexpensive boundary flux correction regains second order accuracy Using FEM Flux Using Boundary Flux 4.5 log(l 2 error) log(sqrt(degrees of freedom)) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 71/77
72 Conjugate Heat Transfer Between a Fluid and a Solid Quantity of interest is the temperature at a small region in the center of the cylinder Flux correction is applied for the computation on the right Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 72/77
73 Additional Problems Additional problems that have been tackled include: Multirate solution methods for ordinary differential equations and parabolic problems IMEX (Implicit-Explicit) solution methods for parabolic problems Explicit time integration methods for ordinary differential equations Finite volume, mixed finite element methods, and mortar mixed finite element methods for elliptic and parabolic problems with coupling through a common boundary Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 73/77
74 Conclusion Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 74/77
75 Conclusions We have developed a powerful analysis framework for a posteriori analysis of multiscale operator decomposition methods It accounts for various sources of error: The solution of each component Information that is passed between components Incomplete iteration Differences between the adjoints to the original problem and a multiscale operator decomposition discretization The analysis typically involves solving auxiliary adjoint problems as a way of quantifying the effect on stability Additional work is required to obtain computable estimates Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 75/77
76 Papers A posteriori - a priori analysis of multiscale operator splitting, D. Estep, V. Ginting, D. Ropp, J. Shadid, and S. Tavener, SINUM, 46 (2008) A posteriori error estimation of approximate boundary fluxes, T. Wildey, S. Tavener, and D. Estep, Comm. Num. Meth. Engin., 24 (2008) A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem, D. Estep, S. Tavener, T. Wildey, SINUM, 46 (2008) A posteriori error analysis of multiscale operator decomposition methods for multiphysics models, D. Estep, V. Carey, V. Ginting, S. Tavener, T. Wildey, J. Physics: Conf. Ser. 125 (2008) A posteriori analysis and adaptive error control for multiscale operator decomposition methods for coupled elliptic systems I: One way coupled systems, V. Carey, D. Estep, and S. Tavener, SINUM 47 (2009) A posteriori error analysis for a transient conjugate heat transfer problem, D. Estep, S. Tavener, T. Wildey, Fin. Elem. Analysis Design 45 (2009) Bridging the Scales in Science and Engineering, Editor J. Fish, Oxford University Press, Chapter 11: Error Estimation for Multiscale Operator Decomposition for Multiphysics Problems, D. Estep, 2010 A posteriori error estimation and adaptive mesh refinement for a multi-discretization operator decomposition approach to fluid-solid heat transfer, D. Estep, S. Tavener, T. Wildey, JCP 229 (2010) A posteriori analysis of multirate numerical methods for ordinary differential equations, SINUM, D. Estep, V. Ginting, S. Tavener, Comput. Meth. Appl. Mech. Eng (2012) Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 76/77
77 Papers A posteriori analysis of an iterative multi-discretization method for reaction-diffusion systems, D. Estep. V. Ginting, J. Hameed, and S. Tavener, Comp. Meth. Appl. Mech. Engin. 267 (2013) A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems, V. Carey, D. Estep, S. Tavener, Int. J. Numer. Meth. Engin. 94 (2013) A-posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, IMA J. Numer. Analysis (2013) A posteriori error estimation for the Lax-Wendroff finite difference scheme, J. B. Collins, D. Estep, and S. Tavener, J. Comput. Appl. Math. 263C (2014) A posteriori error estimates for explicit time integration methods, J. Collins, D. Estep and S. Tavener, BIT (2012), in revision The interaction of iteration error and stability for time dependent linear partial differential equations coupled through an interface, B. Sheehan, D. Estep, S. Tavener, J. Cary, S. Kruger, A. Hakim, A. Pletzer, J. Carlsson, Numer. Alg. (2013), submitted A posteriori error analysis of IMEX time integration schemes for advection-diffusion-reaction equations, J. Chaudry, D. Estep, V. Ginting, J. Shadid, and S. Tavener, (2014), in preparation A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary with non-matching discretizations, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, (2014), in preparation Donald Estep: Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems 77/77
Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems
Adjoint-Fueled Advances in Error Estimation for Multiscale, Multiphysics Systems Donald Estep University Interdisciplinary Research Scholar Department of Statistics Department of Mathematics Center for
More informationDuality, Adjoint Operators, and Uncertainty in a Complex World
Donald Estep, Colorado State University p. 1/196 Duality, Adjoint Operators, and Uncertainty in a Complex World Donald Estep Department of Mathematics Department of Statistics Colorado State University
More informationA posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques
BIT Numer Math DOI 10.1007/s10543-014-0534-9 A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques J. B. Collins D. Estep S.
More informationA posteriori error analysis of multiscale operator decomposition methods for multiphysics models
A posteriori error analysis of multiscale operator decomposition metods for multipysics models D. Estep, V. Carey 2, V. Ginting 3, S. Tavener 2, T. Wildey 4 5 Department of Matematics and Department of
More informationA posteriori error estimation of approximate boundary fluxes
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2008; 24:421 434 Published online 24 May 2007 in Wiley InterScience (www.interscience.wiley.com)..1014 A posteriori error estimation
More informationUtilizing Adjoint-Based Techniques to Improve the Accuracy and Reliability in Uncertainty Quantification
Utilizing Adjoint-Based Techniques to Improve the Accuracy and Reliability in Uncertainty Quantification Tim Wildey Sandia National Laboratories Center for Computing Research (CCR) Collaborators: E. Cyr,
More informationELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR PARABOLIC PROBLEMS
ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR PARABOLIC PROBLEMS CHARALAMBOS MAKRIDAKIS AND RICARDO H. NOCHETTO Abstract. It is known that the energy technique for a posteriori error analysis
More informationTexas at Austin 2 Nazarbayev University. January 15, 2019
Recovery of the Interface Velocity for the Incompressible Flow in Enhanced Velocity Mixed Finite Element Method arxiv:1901.04401v1 [math.na] 14 Jan 2019 Yerlan Amanbek 1,2, Gurpreet Singh 1, and Mary F.
More informationChapter 6. Finite Element Method. Literature: (tiny selection from an enormous number of publications)
Chapter 6 Finite Element Method Literature: (tiny selection from an enormous number of publications) K.J. Bathe, Finite Element procedures, 2nd edition, Pearson 2014 (1043 pages, comprehensive). Available
More informationarxiv: v1 [math.na] 29 Feb 2016
EFFECTIVE IMPLEMENTATION OF THE WEAK GALERKIN FINITE ELEMENT METHODS FOR THE BIHARMONIC EQUATION LIN MU, JUNPING WANG, AND XIU YE Abstract. arxiv:1602.08817v1 [math.na] 29 Feb 2016 The weak Galerkin (WG)
More informationarxiv: v1 [math.na] 6 Nov 2017
Efficient boundary corrected Strang splitting Lukas Einkemmer Martina Moccaldi Alexander Ostermann arxiv:1711.02193v1 [math.na] 6 Nov 2017 Version of November 6, 2017 Abstract Strang splitting is a well
More informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationGENERALIZED GREEN S FUNCTIONS AND THE EFFECTIVE DOMAIN OF INFLUENCE
GENERALIZED GREEN S FUNCTIONS AND THE EFFECTIVE DOMAIN OF INFLUENCE DONALD ESTEP, MICHAEL HOLST, AND MATS LARSON Abstract. One well-known approach to a posteriori analysis of finite element solutions of
More informationNew Multigrid Solver Advances in TOPS
New Multigrid Solver Advances in TOPS R D Falgout 1, J Brannick 2, M Brezina 2, T Manteuffel 2 and S McCormick 2 1 Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O.
More informationTong Sun Department of Mathematics and Statistics Bowling Green State University, Bowling Green, OH
Consistency & Numerical Smoothing Error Estimation An Alternative of the Lax-Richtmyer Theorem Tong Sun Department of Mathematics and Statistics Bowling Green State University, Bowling Green, OH 43403
More informationMulti-Factor Finite Differences
February 17, 2017 Aims and outline Finite differences for more than one direction The θ-method, explicit, implicit, Crank-Nicolson Iterative solution of discretised equations Alternating directions implicit
More informationThe method of lines (MOL) for the diffusion equation
Chapter 1 The method of lines (MOL) for the diffusion equation The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
More informationIndex. higher order methods, 52 nonlinear, 36 with variable coefficients, 34 Burgers equation, 234 BVP, see boundary value problems
Index A-conjugate directions, 83 A-stability, 171 A( )-stability, 171 absolute error, 243 absolute stability, 149 for systems of equations, 154 absorbing boundary conditions, 228 Adams Bashforth methods,
More informationIntroduction to Aspects of Multiscale Modeling as Applied to Porous Media
Introduction to Aspects of Multiscale Modeling as Applied to Porous Media Part IV Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and
More informationA Framework for Analyzing and Constructing Hierarchical-Type A Posteriori Error Estimators
A Framework for Analyzing and Constructing Hierarchical-Type A Posteriori Error Estimators Jeff Ovall University of Kentucky Mathematics www.math.uky.edu/ jovall jovall@ms.uky.edu Kentucky Applied and
More informationarxiv: v1 [math.na] 12 Aug 2013
ADAPTIVE FINITE ELEMENTS FOR SEMILINEAR REACTION-DIFFUSION SYSTEMS ON GROWING DOMAINS CHANDRASEKHAR VENKATARAMAN, OMAR LAKKIS, AND ANOTIDA MADZVAMUSE arxiv:138.449v1 [math.na] 1 Aug 13 Abstract. We propose
More informationAn optimal control problem for a parabolic PDE with control constraints
An optimal control problem for a parabolic PDE with control constraints PhD Summer School on Reduced Basis Methods, Ulm Martin Gubisch University of Konstanz October 7 Martin Gubisch (University of Konstanz)
More informationNUMERICAL METHODS FOR ENGINEERING APPLICATION
NUMERICAL METHODS FOR ENGINEERING APPLICATION Second Edition JOEL H. FERZIGER A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationLocal pointwise a posteriori gradient error bounds for the Stokes equations. Stig Larsson. Heraklion, September 19, 2011 Joint work with A.
Local pointwise a posteriori gradient error bounds for the Stokes equations Stig Larsson Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg Heraklion, September
More informationNumerical Approximation of Phase Field Models
Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School
More informationDiscrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction
Discrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction Problem Jörg-M. Sautter Mathematisches Institut, Universität Düsseldorf, Germany, sautter@am.uni-duesseldorf.de
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationFinite difference methods for the diffusion equation
Finite difference methods for the diffusion equation D150, Tillämpade numeriska metoder II Olof Runborg May 0, 003 These notes summarize a part of the material in Chapter 13 of Iserles. They are based
More informationGoal. Robust A Posteriori Error Estimates for Stabilized Finite Element Discretizations of Non-Stationary Convection-Diffusion Problems.
Robust A Posteriori Error Estimates for Stabilized Finite Element s of Non-Stationary Convection-Diffusion Problems L. Tobiska and R. Verfürth Universität Magdeburg Ruhr-Universität Bochum www.ruhr-uni-bochum.de/num
More informationRecovery-Based A Posteriori Error Estimation
Recovery-Based A Posteriori Error Estimation Zhiqiang Cai Purdue University Department of Mathematics, Purdue University Slide 1, March 2, 2011 Outline Introduction Diffusion Problems Higher Order Elements
More informationA high order adaptive finite element method for solving nonlinear hyperbolic conservation laws
A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws Zhengfu Xu, Jinchao Xu and Chi-Wang Shu 0th April 010 Abstract In this note, we apply the h-adaptive streamline
More informationAdaptive methods for control problems with finite-dimensional control space
Adaptive methods for control problems with finite-dimensional control space Saheed Akindeinde and Daniel Wachsmuth Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy
More informationMultigrid solvers for equations arising in implicit MHD simulations
Multigrid solvers for equations arising in implicit MHD simulations smoothing Finest Grid Mark F. Adams Department of Applied Physics & Applied Mathematics Columbia University Ravi Samtaney PPPL Achi Brandt
More informationSimulating Solid Tumor Growth using Multigrid Algorithms
Simulating Solid Tumor Growth using Multigrid Applied Mathematics, Statistics, and Program Advisor: Doron Levy, PhD Department of Mathematics/CSCAMM University of Maryland, College Park September 30, 2014
More informationResearch Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
Abstract and Applied Analysis Volume 212, Article ID 391918, 11 pages doi:1.1155/212/391918 Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations Chuanjun Chen
More informationStability of flow past a confined cylinder
Stability of flow past a confined cylinder Andrew Cliffe and Simon Tavener University of Nottingham and Colorado State University Stability of flow past a confined cylinder p. 1/60 Flow past a cylinder
More informationAM 205: lecture 14. Last time: Boundary value problems Today: Numerical solution of PDEs
AM 205: lecture 14 Last time: Boundary value problems Today: Numerical solution of PDEs ODE BVPs A more general approach is to formulate a coupled system of equations for the BVP based on a finite difference
More informationUne méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé
Une méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé CMCS/IACS Ecole Polytechnique Federale de Lausanne Erik.Burman@epfl.ch Méthodes Numériques
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 11 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002.
More informationNonparametric density estimation for elliptic problems with random perturbations
Nonparametric density estimation for elliptic problems with random perturbations, DqF Workshop, Stockholm, Sweden, 28--2 p. /2 Nonparametric density estimation for elliptic problems with random perturbations
More informationOn Multigrid for Phase Field
On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004 Synopsis
More informationOutline: 1 Motivation: Domain Decomposition Method 2 3 4
Multiscale Basis Functions for Iterative Domain Decomposition Procedures A. Francisco 1, V. Ginting 2, F. Pereira 3 and J. Rigelo 2 1 Department Mechanical Engineering Federal Fluminense University, Volta
More informationParallel Simulation of Subsurface Fluid Flow
Parallel Simulation of Subsurface Fluid Flow Scientific Achievement A new mortar domain decomposition method was devised to compute accurate velocities of underground fluids efficiently using massively
More informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationSolving PDEs with Multigrid Methods p.1
Solving PDEs with Multigrid Methods Scott MacLachlan maclachl@colorado.edu Department of Applied Mathematics, University of Colorado at Boulder Solving PDEs with Multigrid Methods p.1 Support and Collaboration
More informationSolving PDEs with freefem++
Solving PDEs with freefem++ Tutorials at Basque Center BCA Olivier Pironneau 1 with Frederic Hecht, LJLL-University of Paris VI 1 March 13, 2011 Do not forget That everything about freefem++ is at www.freefem.org
More informationNumerical Methods for Problems with Moving Fronts Orthogonal Collocation on Finite Elements
Electronic Text Provided with the Book Numerical Methods for Problems with Moving Fronts by Bruce A. Finlayson Ravenna Park Publishing, Inc., 635 22nd Ave. N. E., Seattle, WA 985-699 26-524-3375; ravenna@halcyon.com;www.halcyon.com/ravenna
More informationFinite Difference and Finite Element Methods
Finite Difference and Finite Element Methods Georgy Gimel farb COMPSCI 369 Computational Science 1 / 39 1 Finite Differences Difference Equations 3 Finite Difference Methods: Euler FDMs 4 Finite Element
More informationALGEBRAIC FLUX CORRECTION FOR FINITE ELEMENT DISCRETIZATIONS OF COUPLED SYSTEMS
Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2007 M. Papadrakakis, E. Oñate and B. Schrefler (Eds) c CIMNE, Barcelona, 2007 ALGEBRAIC FLUX CORRECTION
More informationKasetsart University Workshop. Multigrid methods: An introduction
Kasetsart University Workshop Multigrid methods: An introduction Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu A copy of these slides is available
More informationMultigrid Methods and their application in CFD
Multigrid Methods and their application in CFD Michael Wurst TU München 16.06.2009 1 Multigrid Methods Definition Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential
More informationNumerical Methods for Engineers and Scientists
Numerical Methods for Engineers and Scientists Second Edition Revised and Expanded Joe D. Hoffman Department of Mechanical Engineering Purdue University West Lafayette, Indiana m MARCEL D E К К E R MARCEL
More informationDiffusion / Parabolic Equations. PHY 688: Numerical Methods for (Astro)Physics
Diffusion / Parabolic Equations Summary of PDEs (so far...) Hyperbolic Think: advection Real, finite speed(s) at which information propagates carries changes in the solution Second-order explicit methods
More informationNon-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions
Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions Wayne McGee and Padmanabhan Seshaiyer Texas Tech University, Mathematics and Statistics (padhu@math.ttu.edu) Summary. In
More informationWeighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods
Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu
More informationNonparametric density estimation for randomly perturbed elliptic problems II: Applications and adaptive modeling
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 29; 8:846 867 Published online 6 January 29 in Wiley InterScience (www.interscience.wiley.com). DOI:.2/nme.2547 Nonparametric
More informationFinite Difference Solution of the Heat Equation
Finite Difference Solution of the Heat Equation Adam Powell 22.091 March 13 15, 2002 In example 4.3 (p. 10) of his lecture notes for March 11, Rodolfo Rosales gives the constant-density heat equation as:
More informationA Two-Grid Stabilization Method for Solving the Steady-State Navier-Stokes Equations
A Two-Grid Stabilization Method for Solving the Steady-State Navier-Stokes Equations Songul Kaya and Béatrice Rivière Abstract We formulate a subgrid eddy viscosity method for solving the steady-state
More informationA Balancing Algorithm for Mortar Methods
A Balancing Algorithm for Mortar Methods Dan Stefanica Baruch College, City University of New York, NY 11, USA Dan Stefanica@baruch.cuny.edu Summary. The balancing methods are hybrid nonoverlapping Schwarz
More informationBasic Aspects of Discretization
Basic Aspects of Discretization Solution Methods Singularity Methods Panel method and VLM Simple, very powerful, can be used on PC Nonlinear flow effects were excluded Direct numerical Methods (Field Methods)
More informationSECOND ORDER TIME DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
Proceedings of ALGORITMY 2009 pp. 1 10 SECOND ORDER TIME DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS MILOSLAV VLASÁK Abstract. We deal with a numerical solution of a scalar
More informationIntroduction to Heat and Mass Transfer. Week 9
Introduction to Heat and Mass Transfer Week 9 補充! Multidimensional Effects Transient problems with heat transfer in two or three dimensions can be considered using the solutions obtained for one dimensional
More informationSpace-time XFEM for two-phase mass transport
Space-time XFEM for two-phase mass transport Space-time XFEM for two-phase mass transport Christoph Lehrenfeld joint work with Arnold Reusken EFEF, Prague, June 5-6th 2015 Christoph Lehrenfeld EFEF, Prague,
More informationThermal Analysis Contents - 1
Thermal Analysis Contents - 1 TABLE OF CONTENTS 1 THERMAL ANALYSIS 1.1 Introduction... 1-1 1.2 Mathematical Model Description... 1-3 1.2.1 Conventions and Definitions... 1-3 1.2.2 Conduction... 1-4 1.2.2.1
More informationWeak Galerkin Finite Element Methods and Applications
Weak Galerkin Finite Element Methods and Applications Lin Mu mul1@ornl.gov Computational and Applied Mathematics Computationa Science and Mathematics Division Oak Ridge National Laboratory Georgia Institute
More informationSimulating Solid Tumor Growth Using Multigrid Algorithms
Simulating Solid Tumor Growth Using Multigrid Algorithms Asia Wyatt Applied Mathematics, Statistics, and Scientific Computation Program Advisor: Doron Levy Department of Mathematics/CSCAMM Abstract In
More informationITERATIVE METHODS FOR NONLINEAR ELLIPTIC EQUATIONS
ITERATIVE METHODS FOR NONLINEAR ELLIPTIC EQUATIONS LONG CHEN In this chapter we discuss iterative methods for solving the finite element discretization of semi-linear elliptic equations of the form: find
More informationDiscontinuous Galerkin Method for interface problem of coupling different order elliptic equations
Discontinuous Galerkin Method for interface problem of coupling different order elliptic equations Igor Mozolevski, Endre Süli Federal University of Santa Catarina, Brazil Oxford University Computing Laboratory,
More informationA Measure-Theoretic Computational Method for Inverse Sensitivity Problems III: Multiple Quantities of Interest
SIAM/ASA J. UNCERTAINTY QUANTIFICATION Vol., pp. 74 c 4 Society for Industrial and Applied Mathematics and American Statistical Association A Measure-Theoretic Computational Method for Inverse Sensitivity
More informationNUMERICAL INVESTIGATION OF BUOY- ANCY DRIVEN FLOWS IN TIGHT LATTICE FUEL BUNDLES
Fifth FreeFem workshop on Generic Solver for PDEs: FreeFem++ and its applications NUMERICAL INVESTIGATION OF BUOY- ANCY DRIVEN FLOWS IN TIGHT LATTICE FUEL BUNDLES Paris, December 12 th, 2013 Giuseppe Pitton
More informationComparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes
Comparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes Do Y. Kwak, 1 JunS.Lee 1 Department of Mathematics, KAIST, Taejon 305-701, Korea Department of Mathematics,
More informationThe CG1-DG2 method for conservation laws
for conservation laws Melanie Bittl 1, Dmitri Kuzmin 1, Roland Becker 2 MoST 2014, Germany 1 Dortmund University of Technology, Germany, 2 University of Pau, France CG1-DG2 Method - Motivation hp-adaptivity
More informationParallel Galerkin Domain Decomposition Procedures for Parabolic Equation on General Domain
Parallel Galerkin Domain Decomposition Procedures for Parabolic Equation on General Domain Keying Ma, 1 Tongjun Sun, 1 Danping Yang 1 School of Mathematics, Shandong University, Jinan 50100, People s Republic
More informationEfficient FEM-multigrid solver for granular material
Efficient FEM-multigrid solver for granular material S. Mandal, A. Ouazzi, S. Turek Chair for Applied Mathematics and Numerics (LSIII), TU Dortmund STW user committee meeting Enschede, 25th September,
More information1. Introduction. We consider the model problem that seeks an unknown function u = u(x) satisfying
A SIMPLE FINITE ELEMENT METHOD FOR LINEAR HYPERBOLIC PROBLEMS LIN MU AND XIU YE Abstract. In this paper, we introduce a simple finite element method for solving first order hyperbolic equations with easy
More informationA PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM
A PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM Hyun Geun LEE 1, Jeong-Whan CHOI 1 and Junseok KIM 1 1) Department of Mathematics, Korea University, Seoul
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationModeling of two-phase flow in fractured porous media on unstructured non-uniform coarse grids
Modeling of two-phase flow in fractured porous media on unstructured non-uniform coarse grids Jørg Espen Aarnes and Vera Louise Hauge SINTEF ICT, Deptartment of Applied Mathematics Applied Mathematics
More informationTime stepping methods
Time stepping methods ATHENS course: Introduction into Finite Elements Delft Institute of Applied Mathematics, TU Delft Matthias Möller (m.moller@tudelft.nl) 19 November 2014 M. Möller (DIAM@TUDelft) Time
More informationFluctuating Hydrodynamics and Direct Simulation Monte Carlo
Fluctuating Hydrodynamics and Direct Simulation Monte Carlo Kaushi Balarishnan Lawrence Bereley Lab John B. Bell Lawrence Bereley Lab Alesandar Donev New Yor University Alejandro L. Garcia San Jose State
More informationPartitioned Methods for Multifield Problems
C Partitioned Methods for Multifield Problems Joachim Rang, 6.7.2016 6.7.2016 Joachim Rang Partitioned Methods for Multifield Problems Seite 1 C One-dimensional piston problem fixed wall Fluid flexible
More informationHIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS
HIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS JASON ALBRIGHT, YEKATERINA EPSHTEYN, AND QING XIA Abstract. Highly-accurate numerical methods that can efficiently
More informationTutorial 2. Introduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes c 2012 Classifying PDEs Looking at the PDE Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu +.. = 0, and its discriminant, B 2
More informationBang bang control of elliptic and parabolic PDEs
1/26 Bang bang control of elliptic and parabolic PDEs Michael Hinze (joint work with Nikolaus von Daniels & Klaus Deckelnick) Jackson, July 24, 2018 2/26 Model problem (P) min J(y, u) = 1 u U ad,y Y ad
More information1 Introduction. J.-L. GUERMOND and L. QUARTAPELLE 1 On incremental projection methods
J.-L. GUERMOND and L. QUARTAPELLE 1 On incremental projection methods 1 Introduction Achieving high order time-accuracy in the approximation of the incompressible Navier Stokes equations by means of fractional-step
More informationA Stable Spectral Difference Method for Triangles
A Stable Spectral Difference Method for Triangles Aravind Balan 1, Georg May 1, and Joachim Schöberl 2 1 AICES Graduate School, RWTH Aachen, Germany 2 Institute for Analysis and Scientific Computing, Vienna
More information1. Introduction. Let the nonlinear set of equations arising in nonlinear transient elasticity be given by. r(u) = 0, (1.1)
PSEUDO-TRANSIENT CONTINUATION FOR NONLINEAR TRANSIENT ELASTICITY MICHAEL W. GEE, C. T. KELLEY, AND R. B. LEHOUCQ Abstract. We show how the method of pseudo-transient continuation can be applied to improve
More informationFinite Element Decompositions for Stable Time Integration of Flow Equations
MAX PLANCK INSTITUT August 13, 2015 Finite Element Decompositions for Stable Time Integration of Flow Equations Jan Heiland, Robert Altmann (TU Berlin) ICIAM 2015 Beijing DYNAMIK KOMPLEXER TECHNISCHER
More informationMarching on the BL equations
Marching on the BL equations Harvey S. H. Lam February 10, 2004 Abstract The assumption is that you are competent in Matlab or Mathematica. White s 4-7 starting on page 275 shows us that all generic boundary
More informationFinite difference method for solving Advection-Diffusion Problem in 1D
Finite difference method for solving Advection-Diffusion Problem in 1D Author : Osei K. Tweneboah MATH 5370: Final Project Outline 1 Advection-Diffusion Problem Stationary Advection-Diffusion Problem in
More informationNumerical Analysis of Evolution Problems in Multiphysics
Thesis for the Degree of Doctor of Philosophy Numerical Analysis of Evolution Problems in Multiphysics Anna Persson Division of Mathematics Department of Mathematical Sciences Chalmers University of Technology
More informationSpectral element agglomerate AMGe
Spectral element agglomerate AMGe T. Chartier 1, R. Falgout 2, V. E. Henson 2, J. E. Jones 4, T. A. Manteuffel 3, S. F. McCormick 3, J. W. Ruge 3, and P. S. Vassilevski 2 1 Department of Mathematics, Davidson
More informationIterative Solvers in the Finite Element Solution of Transient Heat Conduction
Iterative Solvers in the Finite Element Solution of Transient Heat Conduction Mile R. Vuji~i} PhD student Steve G.R. Brown Senior Lecturer Materials Research Centre School of Engineering University of
More informationA Posteriori Error Estimation Techniques for Finite Element Methods. Zhiqiang Cai Purdue University
A Posteriori Error Estimation Techniques for Finite Element Methods Zhiqiang Cai Purdue University Department of Mathematics, Purdue University Slide 1, March 16, 2017 Books Ainsworth & Oden, A posteriori
More informationINTRODUCTION TO MULTIGRID METHODS
INTRODUCTION TO MULTIGRID METHODS LONG CHEN 1. ALGEBRAIC EQUATION OF TWO POINT BOUNDARY VALUE PROBLEM We consider the discretization of Poisson equation in one dimension: (1) u = f, x (0, 1) u(0) = u(1)
More informationA Sobolev trust-region method for numerical solution of the Ginz
A Sobolev trust-region method for numerical solution of the Ginzburg-Landau equations Robert J. Renka Parimah Kazemi Department of Computer Science & Engineering University of North Texas June 6, 2012
More informationContinuous adjoint based error estimation and r-refinement for the active-flux method
Continuous adjoint based error estimation and r-refinement for the active-flux method Kaihua Ding, Krzysztof J. Fidkowski and Philip L. Roe Department of Aerospace Engineering, University of Michigan,
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods
More information