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 Statistics, Colorado State University, Fort Collins, CO 8523 2 Department of Matematics, Colorado State University, Fort Collins, CO 8523 3 Department of Matematics, University of Wyoming, Laramie, WY 827 4 Te Institute for Computational Engineering and Sciences, Te University of Texas, Austin, TX 7872 E-mail: estep@mat.colostate.edu Abstract. Multipysics, multiscale models present significant callenges in terms of computing accurate solutions and for estimating te error in information computed from numerical solutions. In tis paper, we describe recent advances in extending te tecniques of a posteriori error analysis to multiscale operator decomposition solution metods. Wile te particulars of te analysis vary considerably wit te problem, tere are several key ideas underlaying a general approac to treat operator decomposition multiscale metods tat is underdevelopment. We explain tese ideas in te context of tree specific examples.. Multiscale, multipysics problems Multipysics, multiscale models tat couple different pysical processes interacting across a wide range of scales abound in te application areas of te SCIDAC program. Suc models present significant callenges for computing accurate numerical solutions, wic makes it very important to obtain accurate estimates of error in computed information. But, multipysics, multiscale problems also raise significant callenges for error estimation. In tis paper, we describe recent advances in computing a posteriori error estimates for multiscale operator decomposition, wic is a powerful solution metod for multipysics, multiscale problems tat is in widespread use in ig performance computing... Tree examples of multipysics models We briefly describe tree examples of multipysics penomena tat arise in SCIDAC application domains. Tese examples illustrate te myriad ways in wic pysical processes are coupled. Example.. A termal actuator. Tis is a MEMS (microelectronic mecanical switc) device modeled by a system of tree coupled equations, eac representing a distinct pysical 5 Tis work was supported in part by DOE (DE-FG2-4ER2562, DE-FG2-5ER25699, DE-FC2-7ER5499), NASA (NNG4GH63G), NSF (DMS-7832, DMS-7535, DGE-225953, MSPA-CSE-434354, ECCS- 7559), INL (69249), LLNL (B57339) and SNL (PO299784)
process acting on its own scale, (σ u ) =, x Ω, (κ(u 2 ) u 2 ) = σ( u u ), x Ω, (λ tr(e)i + 2µE β(u 2 u 2,ref )I ) =, E = ( () u 3 + u ) 3 /2, x Ω Te first is an electrostatic current equation governing potential u (current J = σ u ), te second is a steady-state energy equation for te governing temperature u 2, and te tird equation is linear elasticity describing te steady-state displacement u 3. Tis is an example of parameter passing, in wic te solution of one component is used to compute te parameters and/or data for anoter component. Note tat u can be calculated independently, wile computing u 2 requires u. Computing u 3 requires u 2 and terefore u. Example.2. Te Brusselator problem. Tis is a model of cemical dynamics [], u t k 2 u = α (β + )u x 2 + u 2 u 2, x (, ), t >, u 2 t k 2 u 2 2 = βu x 2 u 2 u 2, x (, ), t >, u (, t) = u (, t) = α, u 2 (, t) = u 2 (, t) = β/α, t >, u (x, ) = u, (x), u 2 (x, ) = u 2, (x), x (, ), (2) were u and u 2 are te concentrations of species and 2, respectively. Tis problem combines different pysics - in tis case, reaction and diffusion - in one equation. Te generic picture for a reaction-diffusion equation is a relatively fast, destabilizing reaction component interacting wit a relatively slow, stabilizing diffusion component. Tus, te pysical components ave bot different scales and different stability properties. Example.3. Conjugate eat transfer between a fluid and solid object. We consider te flow of a eat-conducting Newtonian fluid past a solid cylinder. Te model consists of te eat equation in te solid and te equations governing te conservation of momentum, mass and energy in te fluid, were we apply te Boussinesq approximation to te momentum equations in te fluid. Te temperature field is advected by te fluid and couples back to te momentum equations troug te buoyancy term. Let Ω F and Ω S be polygonal domains in R 2 wit boundaries Ω F and Ω S intersecting along an interface Γ I = Ω S Ω F. Te complete coupled problem is µ u + ρ (u ) u + p + ρ βt F g = ρ ( + βt ) g, x Ω F, u =, x Ω F, k F T F + ρ c p (u T F ) = Q F, x Ω F, (3) T S = T F, interface x Γ I, k F (n T F ) = k S (n T S ), k S T S = Q S, x Ω S, were we supplement tese equations by imposing Diriclet and Neumann conditions for te velocity field respectively on boundaries Γ u,d and Γ u,n and Diriclet and Neumann conditions for te temperature fields in te fluid and te solid respectively on boundaries Γ TF,D, Γ TF,N, Γ TS,D, and Γ TS,N. Here, ρ and T are reference values for te density and temperature respectively, µ is te molecular viscosity, β is te coefficient of termal expansion, c p is te specific eat, k F and k S are te termal conductivities of te fluid and solid respectively, Q F and Q S are source terms and n is te unit normal vector directed into te fluid. Note tat u is a vector. Tis presents a class of problems were pysics in different pysical domains are coupled across a common boundary.
.2. Callenges and goals for multiscale, multipysics models Callenges arising in te numerical solution of multiscale, multipysics problems include producing accurate numerical solutions efficiently, dealing wit complex stability properties, and transforming information across scales. Anoter complication is te range of applications of multipysics models, including model prediction, sensitivity analysis, and parameter optimization, tat require computation of solutions corresponding to wide range of data and parameters. We expect te solution beavior to vary significantly and te ability to obtain accurate numerical solutions terefore to vary as well. On te oter and, predictive science and engineering ave come to rely on ig performance simulation of complex pysical penomena. In tis context, it is critically important to accurately quantify te numerical error in any computed information. Anoter important goal is determining efficient ways to compute specific information accurately. 2. Multiscale operator decomposition Multiscale operator decomposition is a widely used tecnique for solving multipysics, multiscale models in SCIDAC simulations. Te general approac is to decompose te multipysics and/or multiscale problem into components involving simpler pysics over a relatively limited range of scales, and ten to seek te solution of te entire system troug some sort of iterative procedure involving numerical solutions of te individual components. We illustrate in Fig.. In general, Fully Coupled Multiscale Operator Decomposition Solution Pysics Pysics Pysics 2 Pysics 3 Pys upscale downscale upscale downscale upscale downscale upscale downscale Pysics 2 Pysics 2 Pysics 2 2 Pysics 2 3 Figure. Left: Illustration of a multiscale, multipysics model. Rigt: Illustration of a multiscale operator decomposition solution. different components are solved wit different numerical metods as well as wit different scale discretizations. Reasons tat tis approac is appealing include; Tere is generally a good understanding of ow to solve a broad spectrum of single pysics problems accurately and efficiently. It provides an alternative to accommodating multiple scales in one discretization. It allows for efficient ig performance simulations. However, multiscale operator decomposition generally as significant impact on te accuracy and stability of numerical solutions. 3. Analysis of multiscale operator decomposition In general, multiscale, multipysics problems ave complex stability properties tat defy accurate description via classic a priori analysis. We use duality and adjoint operators to quantify stability a posteriori. We focus on computing a particular quantity of interest determined by a linear functional. We combine tese tools wit variational analysis to produce accurate estimates of te effects of perturbation and error. By numerically solving te adjoint problem, we can compute very accurate error estimates.
Duality and adjoint operators ave a long istory of application in model sensitivity analysis and optimization, dating back to Lagrange, see [2, 3, 4, 5] for example. Te application of tese tools to a posteriori error estimation as a more recent istory, see [6, 7, 8, 9,,, 2, 3]. Our main purpose is to describe recent advances in extending te tecniques of a posteriori error analysis to multiscale operator decomposition solutions of multipysics, multiscale problems. Wile te particulars of te analysis vary considerably wit te problem, tere are several key ideas underlaying a general approac to treat operator decomposition multiscale metods tat is under development, including: We identify auxiliary quantities of interest associated wit information passed between pysical components and solve auxiliary adjoint problems to estimate te error in tose quantities. We deal wit scale differences by introducing projections between discrete spaces used for component solutions and estimate te effects of tose projections. Te standard linearization argument used to define an adjoint operator associated wit error analysis for a nonlinear problem may fail, requiring a new approac. Te adjoint operator associated wit a multiscale operator decomposition solution metod is often significantly different tan te adjoint associated wit te original problem, wic affects te stability of te metod. In practice, solving te adjoint associated wit te original fully-coupled problem may present te same kinds of computational callenges posed by te original problem, so attention must be paid to te solution of te adjoint problem. We explain tese ideas in te context of tree examples. 3.. Multiscale decomposition of triangular systems of elliptic problems We capture te essential features of te termal actuator model in Example. using a oneway coupled system a u + b u + c u = f (x), x Ω, a 2 u 2 + b 2 u 2 + c 2 u 2 = f 2 (x, u, Du ), x Ω, (4) u = u 2 =, x Ω, were a i, b i, c i, f i are smoot functions, wit a, a 2 α > on a bounded domain Ω in R N wit boundary Ω, and α is a constant. We introduce te finite element space S, (Ω) H (Ω), corresponding to a discretization T, of Ω for te first component, and anoter finite element space S,2 (Ω), on a different mes T,2, for te second component. To evaluate integrals involving functions defined on different meses, we introduce projections Π i j from S,i to S,j, e.g. interpolants or an L 2 ortogonal projection. We also define te bilinear forms, A i (u i, v i ) (a i u i, v i )+(b i (x) u i, v i )+(c i u i, v i ), i =, 2. We present te multiscale operator decomposition algoritm in Alg.. Algoritm Multiscale Operator Decomposition for Triangular Systems of Elliptic Equations Construct discretizations T,, T,2 and corresponding spaces S,, S,2 Compute U S, (Ω) satisfying A (U, v ) = (f, v ), for all v S, (Ω). Compute U 2 S,2 (Ω) satisfying A 2 (U 2, v 2 ) = (f 2 (x, Π 2 U, Π 2 DU ), v 2 ), for all v 2 S,2 (Ω). We observe tat any errors made in te solution of te first component affect te solution of te second component. Tis turns out to be a crucial fact for a posteriori error analysis.
Example 3.. We solve a system u = sin(4πx) sin(πy), x Ω u 2 = b u =, x Ω, u = u 2 =, x Ω, b = 2 π ( ) 25 sin(4πx), Ω = [, ] [, ], (5) sin(πx) using a standard piecewise linear, continuous finite element metod to compute te quantity of interest u 2 (.25,.25). We solve for u first and ten solve for u 2 using independent meses. Using uniform meses, evaluating te standard a posteriori error estimate for te second component problem wile ignoring any effect arising from error in te solution of te first component, we estimate te error in te quantity of interest to be.42. However, te true error is.48, so tere is discrepancy of.6 ( 3%) in te estimate. Tis is a consequence of ignoring te error transferred from te solution of te first component. If we adapt te mes for te solution of te second component based on te a posteriori error estimate of te error in tat component wile neglecting te effects of te decomposition, te discrepancy becomes alarmingly worse. For example, we can refine te mes until te estimate of te error in te second component is.. But, we find tat te true error is.2244! See Fig. 2..4.4.6.5.4.6.8.4.4.6.8.5.4.6.8.5.5.5 2.5.4.6.8 Figure 2. Multiscale operator decomposition solutions (u on te left, u 2 in te middle) of te component problems of (5). We solve for u 2 using a mes adapted to control te pointwise error of te numerical solution of u 2 wile ignoring te effect of errors in te approximation of u. On te rigt, we plot te pointwise error of te computed u 2. 3... Description of te a posteriori analysis We seek te error in a quantity of interest given by a functional of e 2 = U 2 u 2. Since we introduce some additional, auxiliary quantities of interest, we denote te primary quantity of interest by (ψ, e) = ( ψ () 2, e 2). We use te adjoint operators A i (φ i, v i ) = (a i φ i, v i ) (div(b i φ i ), v i ) + (c i φ i, v i ), i =, 2. We also use te linearization Lf 2 (u )(u U ) = f 2 u (u s + U ( s)) ds. Noting tat te solution of te first adjoint component is not needed to compute te quantity of interest, we define te primary adjoint problem to be A 2(φ () 2, v 2) = (ψ () 2, v 2), for all v 2 W 2 (Ω). Te error representation formula for te transfer error is ( Df2 (U ) e, Π 2 φ () 2 ) ( + Df2 (U ) e, (I Π 2 )φ () ) 2, (6)
were Π i j is a projection from discretization i to discretization j. Te first term in (6) is te error contribution arising from te transfer wile te second term is significant wen te approximation spaces are different. Te transfer error is a (nominally nonlinear) functional of te error in u, defining an auxiliary quantity of interest. We approximate it by a linear functional and define te auxiliary quantity of interest, ( f2 (u ) f 2 (U ), φ () ) ( 2 Df2 (U ) e, Π 2 φ () ) ( (2) 2 = ψ, e ). We define te corresponding transfer error adjoint problem A (φ (2), v ) = ( ψ (2), v ) for all v W 2 (Ω). (7) Te additional term ( Df 2 (U ) e, (I Π 2 )φ () ) 2 is a linear functional, so we define anoter auxiliary quantity of interest ( (3) ψ, e ) ( = Df2 (U ) e, (I Π 2 )φ () ) 2 and te corresponding adjoint problem Finally, we define te weak residuals A (φ (3), v ) = ( ψ (3), v ) for all v W 2 (Ω). (8) R (U, χ; ν) = (f (ν), χ) A (U, χ), R 2 (U 2, χ; ν) = (f 2 (Π 2 ν), χ) A 2 (U 2, χ). Te final error representation is terefore [4] Teorem 3.2 (ψ, e) = R 2 (U 2, (I Π 2 )φ () 2 ; U ) + R (U, (I Π )(φ (2) + φ (3) )) + ( Π 2 f 2 (U ) f 2 (Π 2 U ), φ () ) ( 2 + (I Π 2 )f 2 (U ), φ () ) (9) 2. We empasize tat evaluating te integrals in (9) is far from trivial. We ave used Monte- Carlo tecniques wit good results, see [4]. Example 3.3. In Example 3., we estimate te contributions to te error reported in tat computation using te relevant portions of (9). To produce te adaptive mes results sown in Fig. 2, we construct te adapted mes using equidistribution based on a bound derived from te first term in (9), i.e. neglecting te terms tat estimate te transfer error. Instead, we consider te system (5) for te quantity of interest equal to te average value of U 2. We begin wit te same initial coarse meses as in Fig. 2, but add te transfer error expression to te mes refinement criterion. Adapting te mes so tat te total error in te quantity of interest for U 2 as error estimates less tan 4 yields te meses sown in Fig. 3. We see tat te first component solve requires significantly more refinement tan te second component. 3.2. Multiscale decomposition of reaction-diffusion problems Operator splitting for reaction-diffusion problems is te classic example of multiscale operator decomposition. We consider u = ɛ u + f(u), x Ω, < t, t suitable boundary conditions, x Ω, < t, () u(, ) = u ( )
.6.4.5.5.4.5.4.6.8.6.8.5.4.6.8 Figure 3. Te adapted meses resulting from te full estimate tat accounts for primary and transfer errors. Te transfer error dominates and te mes for te first component is refined wile te mes for te second component is not. Noneteless, te simulation yields an accurate value for te quantity of interest. It is interesting to compare tis result wit tat sown in Fig. 2. Since te quantity of interest is computed from te second component, intuition migt suggest tat te mes for te second component needs refinement. But, it is te mes for te first component tat is critical for determining accuracy. were Ω R d is a spatial domain and f is a smoot function. Employing te metod of lines, we discretize in space using a continuous, piecewise linear finite element metod wit M elements and obtain te initial value problem: find y R M suc tat ẏ = Ay(t) + F (y(t)), < t T, () y() = y, were A is an l l constant matrix representing a diffusion component and F (y) = (F (y), F 2 (y),, F l (y)) is a vector of nonlinear functions representing a reaction component. We next discretize in time using a discontinuous (dg) Galerkin metod [5, 8]. Wit appropriate coices of quadrature, we can treat standard integration metods. We first discretize [, T ] into = t < t < t 2 < < t N = T wit diffusion time steps t n N n=, t n = t n t n. For eac diffusion step, we coose a (small) time step s n = t n /M n wit s = max n N s n, and te nodes t n = s,n < s,n < < s Mn,n = t n. We define te time intervals I n = [t n, t n ] and I m,n = [s m,n, s m,n ]. See Fig. 4. Te finite element approximate solutions are sougt in Diffusion Integration: Reaction Integration: t t t 2 t 3 t 4 t t t 5 2 t t t 3 4 5 s... s, s M,... s,2 s M2,2 s 2 s 3... s,3 s M3,3... s,4 s M4,4 Figure 4. Discretization of time used for multiscale operator splitting piecewise polynomial spaces for te diffusion and reaction components respectively, V (q d) = U : U In P (q d) (I n ), V (q r) (I n ) = U : U Im,n P (q r) (I m,n ), for n =,, N and, for eac n, m M n. P (q) (I) denotes te space of polynomials in R l of degree q on interval I. We let U n +, denote te left- and rigt-and limits of U at t n and [U] n = U n + Un te jump value of U at t n.
wit Let Ỹ (t) be te piecewise continuous finite element approximation of te operator splitting Ỹ (t) = t n t t n Ỹ n + t t n t n Ỹ n, t n t t n. Te nodal values Ỹn are obtained from te following procedure: Algoritm 2 Multiscale Operator Splitting for Reaction-Diffusion Equations Set Ỹ = y for n =,, N do Set Y r,n = Ỹn for m =,, M n do Compute Y r Im,n P (q r) (I m,n ) satisfying I m,n ( ) Y, W dt + ( [Y r ] m,n, W m + ) = (F (Y r ), W ) dt W P (qr) (I m,n ) (2) I m,n end for Set Y d n = Y r M n,n, compute Y d In P (q d) (I n ) satisfying Set ỹ n = y d (t n ) end for ( ( ) Y d, V ) dt + [Y d ] n, V n + = (AY d, V ) dt V P (qd) (I n ) (3) I n I n Example 3.4. We illustrate te instability of operator splitting applied to te Brusselator problem (2). We apply a standard first order splitting sceme to a space discretization of te Operator Split Solution 2.5.5 -.5.4.6.8 Spatial Location L 2 norm of error - -2-3 t = 6.4 t = 6 t = 32-4 t = 64 t = 8-5 -3-2 - slope Time Step Size Figure 5. Left: Typical instability arising from multiscale operator splitting applied to Brusselator problem. Solution is sown at time 8. Rigt: Plots of te error in te L 2 norm versus time step size at different times. Brusselator model wit 5 discrete points wit α =.6, β = 2, k = k 2 =.25 consisting of te trapezoidal sceme for te diffusion wit time step of.2 and backward Euler sceme for te reaction wit time step of.4. On te left of Fig. 5, we sow a numerical solution tat exibits nonpysical oscillations tat developed after some time. On te rigt, we sow plots of te error versus time steps at different times. Tere is a critical time step above wic te instability develops. Moreover, canging te space discretization does not improve te accuracy. In [6], it is demonstrated tat a finer spatial discretization for a constant time step size leads to significantly more error in te long time solution.
3.2.. Description of te ybrid a posteriori-a priori error analysis An accurate a posteriori estimate of te error must account for te stability effects arising from operator splitting. Te standard approac used to define an adjoint operator for analyzing nonlinear problems based on linearization of a perturbation equation fails. It turns out tat te adjoint operators associated wit te original problem and te multiscale operator decomposition metod are fundamentally different. In te estimate below, tis difference takes te form of residuals between certain adjoint operators. A practical difficulty wit suc a result is tat solving te adjoint for te fully coupled problem poses te same multipysics callenges as solving te original forward problem. We terefore develop a new ybrid a priori - a posteriori estimate tat combines a computable leading order expression obtained using a posteriori arguments wit a provably iger order bound obtained using a priori convergence result. Note tat Ỹ is a consistent numerical solution for an analytic version of te operator splitting and te expression for its error can be estimated using te standard a posteriori error analysis. We let ϑ d define te adjoint solution associated wit te diffusion component (3) and ϑ r define te adjoint solution associated wit te reaction component (2) satisfying respectively ϑ d = A ϑ d (t), t n > t t n, ϑ d (t n ) = ψ n, n = N,,, ϑ r = ( ˆF (y r, Y r )) ϑ r (t), s m,n > t s m,n, ϑ r (s m,n ) = ψm,n, r n = N,,, m = M n,,, wit ψ r M n,n = ϑ d+ n and ψ r m,n = ϑ r m,n for m < M n. Tus ϑ r is continuous across te internal reaction time nodes s m,n, m =,, M n. Here, ˆF (y r, Y r ) = F (sy r + ( s)y r ) ds. Te nonlinearity complicates te analysis because te coice of trajectory around wic to linearize is not clear. We cannot use te standard approac of linearizing te error representation because of te operator splitting. Instead, we assume tat bot te original problem and te operator split version ave a common solution and we linearize eac problem in a neigborood of tis common solution. For example, we assume tat y = is a steady state solution of bot problems, wic can be acieved by assuming tat Homogeneity Assumption: F () =, and we linearize in a region around. In terms of applications to reaction-diffusion problems, tere are matematical reasons for tis assumption and it is satisfied in a great many applications. We can modify te analysis to allow for linearization around any known common solution, see [7]. For n =,, N, we define te tree adjoint problems. Te diffusion problem is simplest because it is linear, ϕ d = A ϕ d (t), t n > t t n, ϕ d (t n ) = ψn. d (4) It is convenient to let Φ d n denote te solution operator, so ϕ d (t n ) = Φ d nψ d n. We require two adjoint problems to treat te reaction component. Te difference between te problems is te function around wic tey are linearized, ϕ r = F (Ỹ ) ϕ r (t), t n > t t n, ϕ r (t n ) = ψ r n. ϕ r 2 = F (Y r ) ϕ r 2 (t), t n > t t n, ϕ r 2 (t n ) = ψ r n. If Φ r n(z) denotes te solution operator for te problem linearized around a function z, ten we ave ϕ r (t n ) = Φ r n(ỹ )ψr n and ϕ r 2 (t n ) = Φ r n(y r )ψ r n. We can now prove [7]. (5)
Teorem 3.5 A ybrid a posteriori - a priori error estimate (ỸN y N, ψ N ) = were + E = 2 t n N n= M n n= m= + N ( ) ( Y d AY d, ϑ d Πϑ d ) dt + ([Y d ] n, ϑ d+ n Πϑ d+ n ) I n ( ) ( Y r F (Y r ), ϑ r Πϑ r ) dt + ([Y r ] m,n, ϑ r+ m,n Πϑr+ m,n ) I m,n N (Ỹn, (E + E 2 )ψ n ) + O( t qd+2 ) + O( t s qr+ ), n= (A F(Ỹ ) F(Ỹ )A ), F(Ỹ ) = I n F (Ỹ ) dt, E 2 = ( ) Φ r n(ỹ ) Φr n(y r ) Φ d n. Te first expression on te rigt is te error introduced by te numerical solution of te diffusion component wile te second expression is te error introduced by te numerical solution of te reaction component. Te tird expression on te rigt measures te effects of operator splitting, were E is a leading order estimate for te effects of operator splitting wile E 2 accounts for issues arising from te differences in linearizing around te global computed solution as opposed to te solution of te reaction component. Bot of tese quantities are scaled by te solution itself, so tat tese effects become negligible wen te solution approaces zero. Finally, te remaining terms represent bounds on terms tat are not computable but are iger order. In practice, we neglect tose terms wen computing an estimate. Example 3.6. We consider te Brusselator problem (2) wit α = 2, β = 5.45, k =.8, k 2 =.4 and initial conditions u (x, ) = α +. sin(πx) and u 2 (x, ) = β/α +. sin(πx), wic yields an oscillatory solution. In tis case, te reaction is very mildly unstable, wit at most polynomial rate accumulation of perturbations as time passes. We use a 32 node spatial finite element discretization, resulting in an differential equation system wit dimension 62. We note tat in original form, te reaction terms do not satisfy te requirement F () = so we linearize around te steady state solution c wit wit c i = α for i =,, N e and c i = β/α for i = N e,, 2N e 2, so tat F (c) =. Fig. 6 compares te errors computed using t =. and M = reaction time steps to te ybrid a posteriori error estimates. We sow results for [, 2], wen te solution is still in a transient stage, and at T = 4 wen te solution as become periodic. All te results sow tat te exact and estimated errors are in remarkable agreement. 3.3. Multiscale decomposition of a fluid-solid conjugate eat transfer problem We next consider te multiscale decomposition solution of te eat transfer problem described in Example.. Te weak formulation of (3) consists of computing u V F, p L 2 (Ω F ), T F W F and T S W S suc tat a (u, v) + c (u, u, v) + b(v, p) + d(t F, v) = (f, v), b(u, q) =, (6) a 2 (T F, w F ) + c 2 (u, T F, w F ) + a 3 (T S, w S ) = (Q F, w F ) + (Q S, w S ), for all v V F,, q L 2 (Ω F ), w F W F, and w S W S,, were
Error and Estimate.2. -. Species Species 2 -.2.4.6.8 Component Error and Estimate.2. -. Species Species 2 -.2.5.5 2 Time Error and Estimate.3. -.2 Species Species 2 -.4.4.6.8 Component Figure 6. Brusselator results. Left: Comparison of errors against te spatial location at T = 2. Middle: Time istory of errors at te midpoint location on [, 2]. Rigt: Comparison of errors against te spatial location at T = 4. Te dotted line is te exact error and te (+) is te estimated error and a (u, v) = Ω F µ( u : v) dx, a 2 (T F, w F ) = Ω F k F ( T F w F ) dx, a 3 (T S, w S ) = Ω S k S ( T S w S ) dx, b(v, q) = Ω F ( v)q dx, c (u, v, z) = Ω F ρ (u ) v z dx, c 2 (u, T, w) = Ω F ρ c p (u T ) w dx, d(t, v) = Ω F ρ βt g v dx, f = ρ ( + βt ) g, V F = v H (Ω F ) v = g u,d on Γ u,d, VF, = v V F v = on Γ u,d, W F = w H (Ω F ) w = g TF,D on Γ TF,D, WF, = w W F w = on Γ TF,D, W S = w H (Ω S ) w = g TS,D on Γ TS,D, WS, = w W S w = on Γ TS,D. To discretize, we construct independent locally-quasi-uniform triangulations T F, and T S, of Ω F and Ω S respectively. We use te piecewise polynomial spaces V F = v V F v continuous on Ω F, v i P 2 (K) for all K τ F,, Z = z Z z continuous on Ω F, z P (K) for all K τ F,, W F = w W F w continuous on Ω F, w P 2 (K) for all K τ F,, W S = w W S w continuous on Ω S, w P 2 (K) for all K τ S,, and te associated subspaces VF, = v V v = on Γ u,d, WF, = w WF w = on Γ TF,D, WS, = w WS w = on Γ TS,D and w = on Γ I, were P q (K) denotes te space of polynomials of degree q on an element K. Note tat WS, is different tan WF, in an important way since Γ T S,D does not include Γ I. We let π V, π WF, π WS, and π Z be projections into VF, W F, W S and Z respectively. We also use π WF and π WS to denote projections into WF and W S respectively along te interface Γ I. To compute a stable solution of te fluid equations, we coose VF and Z to be te Taylor- Hood finite element pair satisfying te discrete inf-sup condition. In general, te convergence of te iteration defined by tis Algoritm depends on te values of k S and k F along te interface and te geometry of eac region. Often, a relaxation is used to elp improve convergence properties, wic affects te analysis [8, 9].
Algoritm 3 Multiscale Decomposition Metod for Conjugate Heat Transfer k = wile ( T k S π S T k F Γ I > T OL) do k = k+ Compute T k S, W S Compute u k V F, pk k suc tat T S, = π W S T k F, along te interface Γ I and a 3 (T k S,, w) = (Q S, w), w W S,, (7) Z and T k F, W F suc tat a (u k, v) + c (u k, uk k, v) + b(v, pk ) + d(t F,, v) = (f, v), v V F,, b(u k, q) =, q Z, (8) a 2 (T k F,, w) + c 2(u k, T k F,, w) = (Q F, w) (k S (n T k S, ), w) Γ I, w WF,. end wile 3.3.. Description of an a posteriori error analysis We define te errors e u = u u k e p = p p k, e T F = T F T k F,, and e T S = T S T k S,. Te adjoint problem for te quantity of interest (ψ, e) = (ψ u, e u ) + (ψ p, e p ) + (ψ TF, e TF ) + (ψ TS, e TS ) is µ φ + c (φ) + z + c 2u (θ F ) = ψ u, x Ω F, φ = ψ p, x Ω F, k F θ F + c 2T (θ F ) + ρ β(g φ) = ψ TF, x Ω F, θ F = θ S, x Γ I, k F (n θ F ) = k S (n θ S ), k S θ S = ψ TS, x Ω S, Here, we ave used te linearizations φ =, x Γ u,d, µ φ n =, x Γ u,n, θ F =, x Γ TF,D, k F (n θ F ) =, x Γ TF,N, θ S =, x Γ TS,D, k S (n θ S ) =, x Γ TS,N. c (φ) = 2 ρ (u + u ) φ 2 ρ (u + u ) φ 2 ρ ( (u + u )) φ, c 2u(θ) = 2 ρ c p (T + T ) θ, c 2T (θ) = 2 ρ c p (u + u ) θ 2 ρ c p ( (u + u )) θ. We solve (9) numerically using an iterative operator decomposition approac as for te forward problem independently of te forward iterations. In [8, 9], we derive estimates tat only require adjoint solutions of te two component problems. To write out te a posteriori error representation, we introduce an additional projection π W S : H 2 W S, defined suc tat for any node x i. π W S θ S (x i ) = π WS θ S (x i ) for x i / Γ I, and oterwise. We can now prove [9],, (9)
Teorem 3.7 (ψ, e) = (f, φ π V φ) a (u k, φ π V φ ) c (u k, uk, φ π V φ) b(φ π V φ, p ) d(t k F,, φ π V φ) b(u k, z π Zz) (2) + (Q F, θ F π WF θ F ) a 2 (T k F,, θ F π WF θ F ) c 2 (u k, T k F,, θ F π WF θ F ) + (Q S, θ S π WS θ S ) a 3 (T k S,, θ S π WS θ S ) (2) + ( T k S, π ST k F,, k S(n θ S ) ) + ( π Γ I S T k F, T k F,, k S(n θ S ) ) (22) Γ I + ( k S (n T k S, ), π ) W F θ F + ( ) ( k Q Γ I S, π WS θ S a3 T S,, π ) W S θ S. (23) Te contributions to te error are Equations (2)-(2) represents te contribution of te discretization error arising from eac component solve. Equation (22) represents te contribution from te iteration. Te first term in (23) represents contribution of te transfer error wile te remaining terms represent te contribution arising from projections between two different discretizations. Example 3.8. For te flow past a cylinder, we solve te steady non-dimensionalized Boussinesq equations in te fluid domain and te non-dimensional eat equation in te solid domain. To simulate te flow of water past a cylinder made from stainless steel, we set te dimensionless constants P r = 6.6 and k R = 3, and coose te inflow velocity and te temperature gradient so tat Re = 75, P e = 495, F r =., Ra = 5. Te temperature gradient is imposed by setting different temperatures along te top and bottom boundaries, wit a linear temperature gradient on te inflow boundary, and an adiabatic condition on te outflow boundary. We sow results for two quantities of interest. Te first is te temperature in a small region in te wake, located approximately one cannel widt downstream of te center of te cylinder and /4 of a cannel widt below te upper wall. Te second is temperature at a small region in te center of te cylinder. In eac case, we base adaptivity on an element tolerance of 8. We sow te final adaptive meses for te flow and solid in Fig. 7. For te first quantity of interest, te flow mes is most refined near te region of interest and upstream of te region of interest, locating more elements between te cylinder and te top wall tan te cylinder and te bottom wall since te flow advecting eat to te region of interest passes above rater tan below te cylinder. For te solid, te mes is igly refined along te top in order to increase te accuracy of te normal derivative tat is computed in te solid and used as a boundary condition in te fluid computation. Evidently, te normal derivatives elsewere on te interface ave less of an influence on te first quantity of interest. For te second quantity of interest, te mes is igly refined upstream of te cylinder. We note tat te refinement downstream of te cylinder corresponds closely to te recirculation region, and te mes refinement is sligtly asymmetric about te midplane of te cannel due to te asymmetric initial mes. Te mes in te solid is refined uniformly near te boundary, reflecting te fact tat te error in te finite element flux makes a significant contribution to te error in te quantity of interest. 3.3.2. Loss of order and flux correction Te meses sown in Fig. 7 are igly refined near te interface. Tis reflects te fact tat tere is significant error in te numerical flux passed between te components. It turns out tat tis pollutes te entire computation, so tat overall te metod loses an entire order of accuracy.
2 2 4 6 2 2 4 6........ Figure 7. Upper: Final adaptive meses in te fluid and solid wen te quantity of interest is te temperature in a small region in te wake above te cylinder. Lower: Final adaptive meses in te fluid and solid wen te quantity of interest is te temperature in a small region in te center of te solid. One way to compensate for te loss of order is by refining te mes locally near te interface. Anoter way is to compute te particular information, in tis case te flux on te interface, more accurately. It turns out tat we can adapt a post-processing tecnique called flux correction developed originally by Weeler [2] and Carey [2, 22] to recover boundary flux values wit increased accuracy. We denote te set of elements in τ S, tat intersect te interface boundary by τ Γ I S, = K τ S, K Γ I, and consider te corresponding finite element space Σ = v P 2 (K) wit K τ Γ I S,, v(η i) = if η i / Γ I, were η i denotes te nodes of element K. Te degrees of freedom correspond to te nodes on te boundary. We compute σ k Σ satisfying ( σ k, v ) Γ I = (Q S, v) a 3 (T k S,, v), for all v Σ, were T k S, solves (7). Since te dimension of te problem scales wit te number of nodes on a boundary, it is relatively inexpensive to solve. In te Algoritm, (8) is replaced by: Compute u k VF, pk Z and T k F, W F suc tat a (u k, v) + c (u k, uk k, v) + b(v, pk ) + d(t F,, v) = (f, v), b(u k, q) =, a 2 (T k F,, w) + c 2(u k, T k F,, w) = (Q F, w) ( σ k, w ), v V Γ I F,, q Z, w WF,. (24) It turns out tat using te recovered boundary flux leads to a cancelation of te transfer error term in te error representation formula (23), wic is te source of te loss of order. We can prove tat using te recovered flux recovers te expected cubic order of convergence, see [9]. Example 3.9. Te recovered accuracy is easily demonstrated by considering te adapted meses produced by te modified algoritm. We sow te final adaptive meses for te solid in Fig. 8. Tere is no mes refinement near te boundaries, indicating tat te flux error is no longer dominant. 4. Conclusion Multipysics, multiscale models present significant callenges in terms of computing accurate solutions and for estimating te error in information computed from numerical solutions. In tis
........ Figure 8. Left: Final adaptive mes in te solid wen te quantity of interest is te temperature in a small region in te wake above te cylinder. Rigt: Final adaptive mes in te solid wen te quantity of interest is te temperature in a small region in te center of te solid. paper, we describe recent advances in extending te tecniques of a posteriori error analysis to multiscale operator decomposition solution metods. Wile te particulars of te analysis vary considerably wit te problem, tere are several key ideas underlaying a general approac to treat operator decomposition multiscale metods tat is under development. We explain tese ideas in te context of tree specific examples. In tis presentation, we ave minimized te effects arising from te solution of nonlinear and/or fully coupled systems by carefully coosing te models and results tat are discussed. Referring back to Fig., we generally expect multiscale operator decomposition to require a number of iterations between te pysical components. Tis raises additional issues tat are discussed in [8, 9, 23, 7, 24]. References [] Prigogine I and Lefever R 968 J. Cem. Pys 48 695 7 [2] Marcuk G I, Agoskov V I and Sutyaev V P 996 Adjoint equations and perturbation algoritms in nonlinear problems (Boca Raton, FL: CRC Press) ISBN -8493-287-3 [3] Lanczos C 997 Linear Differential Operators (Dover Publications) [4] Cacuci D 997 Sensitivity and Uncertainty Analysis: Teory vol I (Capman & Hall/CRC) [5] Marcuk G I 995 Adjoint equations and analysis of complex systems (Kluwer) [6] Eriksson K, Estep D, Hansbo P and Jonson C 996 Computational differential equations (Cambridge: Cambridge University Press) ISBN -52-5632-7; -52-56738-6 [7] Eriksson K, Estep D, Hansbo P and Jonson C 995 Acta numerica, 995 Acta Numer. (Cambridge: Cambridge Univ. Press) pp 5 58 [8] Estep D, Larson M G and Williams R D 2 Mem. Amer. Mat. Soc. 46 viii+9 ISSN 65-9266 [9] Becker R and Rannacer R 2 Acta Numerica 2 [] Giles M and Süli E 22 Acta Numerica 45 236 [] Bangert W and Rannacer R 23 Adaptive Finite Element Metods for Differential Equations (Birkauser Verlag) [2] Parascivoiu M, Peraire J and Patera A 997 Comput. Metods Appl. Mec. Engrg. 5 289 32 [3] Bart T J 24 A-Posteriori Error Estimation and Mes Adaptivity for Finite Volume and Finite Element Metods (Lecture Notes in Computational Science and Engineering vol 4) (New York: Springer) [4] Carey V, Estep D and Tavener S 26 SINUM To appear [5] Estep D and Stuart A M 22 Mat. Comp. 7 75 3 (electronic) ISSN 25-578 [6] Ropp D L, Sadid J N and Ober C C 24 J. Comput. Pys. 94 544 574 ISSN 2-999 [7] Ginting V, Estep D, Sadid J and Tavener S 26 SINUM To appear [8] Estep D, Tavener S and Wildey T 26 SINUM To appear [9] Estep D, Tavener S and Wildey T 28 JCP Submitted [2] Weeler M 974 Numer. Mat 2 99 9 [2] GF Carey SS Cow M S 985 Comp. Met. in Applied Mec. and Engr. 5 7 2 [22] Carey G 982 Comp. Met. in Applied Mec. and Engr. 35 4 [23] Carey V, Estep D and Tavener S 28 In preparation [24] Ginting V, Estep D and Tavener S 28 IMA J. Numer. Analysis Submitted