Matrix-equation-based strategies for convection-diusion equations

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Matrix-equation-based strategies for convection-diusion equations Davide Palitta and Valeria Simoncini Dipartimento di Matematica, Università di Bologna davide.palitta3@unibo.it CIME-EMS Summer School 2015 Exploiting Hidden Structure in Matrix Computations. Algorithms and Applications Cetraro, 22-26 June 2015 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 1 / 23

The convection-diusion equation The convection-diusion equation Convection-diusion equation: ɛ u + w u = f, in Ω R d with d = 2, 3, ɛ > 0 (viscosity parameter), w R d (convection vector) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 2 / 23

The convection-diusion equation The convection-diusion equation Convection-diusion equation: ɛ u + w u = f, in Ω R d with d = 2, 3, ɛ > 0 (viscosity parameter), w R d (convection vector) Dominant Convection w ɛ Incompressible ow div(w) = 0 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 2 / 23

The convection-diusion equation The convection-diusion equation Convection-diusion equation: ɛ u + w u = f, in Ω R d with d = 2, 3, ɛ > 0 (viscosity parameter), w R d (convection vector) Dominant Convection w ɛ Incompressible ow div(w) = 0 Further assumptions: w separable coecients, e.g., in 2D w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) Ω is a rectangle (parallelepipedal) domain Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 2 / 23

The convection-diusion equation Discretization phase Discretizing by FEM or FD, we get the nonsymmetric linear system Au = f, where A R N N Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 3 / 23

The convection-diusion equation Discretization phase Discretizing by FEM or FD, we get the nonsymmetric linear system Au = f, where A R N N REMARK: The discretization step is crucial to avoid spurious oscillations in the numerical solution: Rene the meshsize h huge increase of N Dierent strategies: Articial diusion SUPG... Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 3 / 23

A matrix-equation-based strategy A matrix-equation-based strategy AIM: Preconditioning the very large and sparse nonsymmetric linear system Au = f, where A R N N with a simplied matrix version of the discretized dierential operator Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 4 / 23

A matrix-equation-based strategy A matrix oriented formulation Poisson equation u xx u yy = f, in Ω = (0, 1) 2 (w/ zero Dirichlet b.c.) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 5 / 23

A matrix-equation-based strategy A matrix oriented formulation Poisson equation u xx u yy = f, in Ω = (0, 1) 2 (w/ zero Dirichlet b.c.) Discretize Ω with a uniform mesh Ω h with nodes (x i, y j ), i, j = 1,..., n 1. Dene T, F R (n 1) (n 1) such that T = 1 h 2 tridiag( 1, 2, 1) and F i,j = f (x i, y j ) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 5 / 23

A matrix-equation-based strategy A matrix oriented formulation Poisson equation u xx u yy = f, in Ω = (0, 1) 2 (w/ zero Dirichlet b.c.) Discretize Ω with a uniform mesh Ω h with nodes (x i, y j ), i, j = 1,..., n 1. Dene T, F R (n 1) (n 1) such that T = 1 h 2 tridiag( 1, 2, 1) and F i,j = f (x i, y j ) Centered FD leads to the symmetric linear system Au = f, f = vec(f ) where A = T I n 1 + I n 1 T R (n 1)2 (n 1) 2 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 5 / 23

A matrix-equation-based strategy A matrix oriented formulation U i,j u(x i, y j ) at interior node (x i, y j ) For each i, j = 1,..., n 1, u xx (x i, y j ) U i 1,j 2U i,j + U i+1,j = 1 [1, 2, 1] h 2 h2 U i 1,j U i,j U i+1,j and u yy (x i, y j ) U i,j 1 2U i,j + U i,j+1 h 2 = 1 1 h 2 [U i,j 1, U i,j, U i,j+1] 2 1 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 6 / 23

A matrix-equation-based strategy A matrix oriented formulation U i,j u(x i, y j ) at interior node (x i, y j ) For each i, j = 1,..., n 1, u xx (x i, y j ) U i 1,j 2U i,j + U i+1,j = 1 [1, 2, 1] h 2 h2 U i 1,j U i,j U i+1,j and u yy (x i, y j ) U i,j 1 2U i,j + U i,j+1 h 2 = 1 1 h 2 [U i,j 1, U i,j, U i,j+1] 2 1 TU + UT = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 6 / 23

A matrix-equation-based strategy A matrix oriented formulation, ɛ u + w u = f Proposition. (2D case) Assume w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)). Let (x i, y j ) Ω h, i, j = 1,..., n 1, and set, for k = 1, 2, Φ k = diag(φ k (x 1 ),..., φ k (x n 1 )) Ψ k = diag(ψ k (y 1 ),..., ψ k (y n 1 )) Then, the centered FD discretization of the continuous operator leads to the following operator: L : u ɛ u + φ 1 (x)ψ 1 (y)u x + φ 2 (x)ψ 2 (y)u y L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) where B = 1 tridiag( 1, 0, 1) R(n 1) (n 1) 2h Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 7 / 23

A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 8 / 23

A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Ψ 1 = Φ 2 = I Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 8 / 23

A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Ψ 1 = Φ 2 = I ɛt U + ɛut + (Φ 1 B)U + U(B T Ψ 2 ) = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 8 / 23

A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Ψ 1 = Φ 2 = I ɛt U + ɛut + (Φ 1 B)U + U(B T Ψ 2 ) = F (ɛt + Φ 1 B)U + U(ɛT + B T Ψ 2 ) = F This is a Sylvester equation that can be explicitly and eciently solved! Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 8 / 23

The new preconditioner The general case: preconditioning strategy w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 9 / 23

The new preconditioner The general case: preconditioning strategy w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) Approximating Ψ 1 ψ 1 I, Φ 2 φ 2 I, where, e.g., ψ 1, φ 2 mean values Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 9 / 23

The new preconditioner The general case: preconditioning strategy w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) Approximating Ψ 1 ψ 1 I, Φ 2 φ 2 I, where, e.g., ψ 1, φ 2 mean values P : Y (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + φ 2 B T Ψ 2 ) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 9 / 23

Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 10 / 23

Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. At each iteration k: ṽ k = P 1 v k R (n 1)2 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 10 / 23

Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. At each iteration k: ṽ k = P 1 v k R (n 1)2 How to compute it? Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 10 / 23

Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. At each iteration k: ṽ k = P 1 v k R (n 1)2 How to compute it? 1 Compute G k R (n 1) (n 1) s.t. v k = vec(g k ) 2 Solve (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ2 Ψ 2 ) = G k 3 Compute ṽ k = vec(y) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 10 / 23

Implementation details Implementation details Sylvester equation: (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ 2 Ψ 2 ) = G k Solver: KPIK [V. Simoncini, 2007], [T. Breiten, V. Simoncini, M. Stoll, 2014]. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 11 / 23

Implementation details Implementation details Sylvester equation: (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ 2 Ψ 2 ) = G k Solver: KPIK [V. Simoncini, 2007], [T. Breiten, V. Simoncini, M. Stoll, 2014]. Observations: Inner: Inexact method Outer: FGMRES rhs must be low rank: Truncated SVD of G k Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 11 / 23

Implementation details Implementation details Sylvester equation: (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ 2 Ψ 2 ) = G k Solver: KPIK [V. Simoncini, 2007], [T. Breiten, V. Simoncini, M. Stoll, 2014]. Observations: Inner: Inexact method Outer: FGMRES rhs must be low rank: Truncated SVD of G k Three parameters: 1 inner tolerance: tol_inner= 10 4 2 truncation tolerance: tol_truncation= 10 2 3 maximum rank allowed: r_max= 10 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 11 / 23

3D case 3D case To x ideas: Ω = (0, 1) 3. U (k) i,j u(x i, y j, z k ), and dene the tall matrix U := where e j = I (:, j). U (1). U (n 1) n 1 = k=1 (e k U (k) ) R (n 1)2 (n 1) Other orderings are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 12 / 23

3D case 3D case: L : u ɛ u + w 1 u x + w 2 u y + w 3 u z Proposition. (3D case) Assume w = (w1, w2, w3) s.t. w1 = φ 1(x)ψ 1(y)υ 1(z) w2 = φ 2(x)ψ 2(y)υ 2(z) w3 = φ 3(x)ψ 3(y)υ 3(z) Let (x i, y j, z k ) Ω h, i, j, k = 1,..., n 1, and set, for l = 1, 2, 3, Φ l = diag(φ l (x1),..., φ l (x n 1)) Ψ l = diag(ψ l (y1),..., ψ l (y n 1)) Υ l = diag(υ l (z1),..., υ l (z n 1)) Then, the centered FD discretization leads to the following operator: L h : U (I ɛt )U + ɛut + (ɛt I )U + (Υ 1 Φ 1 B)UΨ 1 + (Υ 2 Φ 2 )UB T Ψ 2 + [(Υ 3 B) Φ 3 ]UΨ 3 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 13 / 23

Numerical experiments Numerical experiments Competitors on Au = f: GMRES+MI20 FGMRES+AGMG REMARK: our preconditioner/solver is implemented in interpreted Matlab functions while both MI20 and AGMG are fortran90 compiled codes provided with mex les control.one_pass_coarsen= 1, [J. Boyle, M.D. Mihajlovi c, J.A. Scott, 2010] all default parameters, [Y. Notay, 2010] Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 14 / 23

Numerical experiments Test 1. Explicit matrix solution (2D) Consider ɛ u+w u = 0, in Ω = (0, 1) 2 ( ) w = 1 + 1 4 (x + 1)2, 0 Dirichlet b.c.: { u(x, 0) = 1 x [0, 1], u(x, 1) = 0 x [0, 1]. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 15 / 23

Numerical experiments Test 1. Explicit matrix solution (2D) Consider ɛ u+w u = 0, in Ω = (0, 1) 2 ( ) w = 1 + 1 4 (x + 1)2, 0 Dirichlet b.c.: { u(x, 0) = 1 x [0, 1], u(x, 1) = 0 x [0, 1]. (ɛt + Φ 1 B)U + ɛut = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 15 / 23

Numerical experiments Test 1. Explicit matrix solution (2D) ɛ n x FGMRES+AGMG GMRES+MI20 KPIK time (# its) time (# its) time (# its) 0.0333 129 0.1649 (13) 0.2843 ( 7) 0.2131 (24) 0.0333 257 0.3874 (15) 0.5715 ( 8) 0.2817 (32) 0.0333 1025 11.4918 (20) 8.4540 ( 8) 1.5002 (44) 0.0333 1200 12.5441 (17) 8.7843 ( 7) 1.9722 (46) 0.0167 129 0.2150 (14) 0.2750 ( 7) 0.1638 (22) 0.0167 257 0.4356 (14) 0.6533 ( 9) 0.3628 (32) 0.0167 513 2.0712 (15) 2.2171 ( 9) 0.6324 (38) 0.0167 1025 11.5428 (18) 8.0454 ( 8) 2.8454 (64) 0.0167 1200 13.2109 (18) 9.5501 ( 8) 2.1961 (52) 0.0083 129 0.1501 (14) 0.2685 (10) 0.1394 (22) 0.0083 257 0.3651 (15) 0.5871 ( 8) 0.2885 (34) 0.0083 513 1.6615 (14) 2.1814 (10) 0.5439 (42) 0.0083 1025 10.0859 (18) 10.6729 (11) 2.7800 (66) 0.0083 1200 14.4866 (18) 11.0856 ( 9) 2.8459 (58) Table : Test 1 (2D). Performance achieved as the viscosity and mesh parameter vary. tol_outer= 10 8. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 16 / 23

Numerical experiments Test 2. Preconditioning strategy (2D) Consider ɛ u + w u = 0, in Ω = (0, 1) 2 w = (y(1 (2x + 1) 2 ), 2(2x + 1)(1 y 2 )) Zero Dirichlet b.c. except for the side y = 0: { u(x, 0) = 1 + tanh[10 + 20(2x 1)], 0 x 0.5 u(x, 0) = 2, 0.5 < x 1 slightly modication of Example V in [H.C. Elman, A. Ramage, 2002]. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 17 / 23

Numerical experiments Test 2. Preconditioning strategy (2D) ɛ n x FGMRES+AGMG GMRES+MI20 FGMRES+KPIK time (# its) time (# its) time (# its) 0.1000 128 0.1081 (11) 0.1989 ( 4) 0.5743 ( 8) 0.1000 256 0.3335 (14) 0.3812 ( 4) 0.5351 ( 7) 0.1000 512 1.0773 (11) 1.1731 ( 4) 1.0543 ( 7) 0.1000 1024 9.2493 (17) 4.3287 ( 4) 2.6372 ( 6) 0.1000 2048 52.0430 (15) 19.6757 ( 4) 16.7394 ( 5) 0.0500 128 0.0936 ( 9) 0.2269 ( 4) 0.5168 (10) 0.0500 256 0.2897 (12) 0.3862 ( 4) 0.6455 ( 9) 0.0500 512 1.2603 (11) 1.2380 ( 4) 1.1769 ( 8) 0.0500 1024 10.3623 (18) 4.3345 ( 4) 3.0812 ( 7) 0.0500 2048 60.5041 (17) 20.4056 ( 4) 14.9237 ( 6) 0.0333 128 0.0882 ( 8) 0.2368 ( 4) 0.6428 (11) 0.0333 256 0.2181 ( 9) 0.4218 ( 4) 0.8149 (11) 0.0333 512 1.5849 (14) 1.1977 ( 4) 1.3786 ( 9) 0.0333 1024 5.4624 (12) 4.4130 ( 4) 3.7214 ( 8) 0.0333 2048 120.9686 (23) 20.1120 ( 4) 17.9188 ( 7) Table : Test 2 (2D). Performance achieved as the viscosity and mesh parameter vary. tol_outer= 10 6. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 18 / 23

Numerical experiments Test 3. Explicit matrix solution (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (x sin x, y cos y, e z2 1 ) Zero Dirichlet b.c. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 19 / 23

Numerical experiments Test 3. Explicit matrix solution (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (x sin x, y cos y, e z2 1 ) Zero Dirichlet b.c. [I (ɛt + Φ 1 B) + (ɛt + Ψ 2 B) T I ] U + U (ɛt + BΥ 3 ) = 11 T where 1 is the vector of all ones REMARK: low rank rhs in the matrix equation is crucial for explicit solution! Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 19 / 23

Numerical experiments Test 3. Explicit matrix solution (3D) ɛ n x FGMRES+AGMG GMRES+MI20 KPIK time (# its) time (# its) time (# its) 0.0050 60 1.1250 (14) 1.8022 ( 7) 0.1734 (18) 0.0050 70 2.0385 (14) 3.4253 ( 7) 0.2326 (20) 0.0050 80 3.4803 (14) 4.4297 ( 7) 0.3583 (20) 0.0050 90 5.7324 (15) 6.8705 ( 7) 0.4999 (22) 0.0050 100 8.0207 (15) 9.7207 ( 7) 0.5677 (22) 0.0010 60 1.3011 (14) 1.7854 ( 7) 0.2386 (18) 0.0010 70 1.9509 (14) 2.7829 ( 7) 0.2346 (20) 0.0010 80 3.5291 (14) 4.6576 ( 7) 0.4096 (20) 0.0010 90 5.1344 (14) 6.8176 ( 7) 0.4253 (22) 0.0010 100 7.6815 (14) 9.4935 ( 7) 0.5446 (22) 0.0005 60 1.2560 (14) 1.7341 ( 6) 0.2314 (18) 0.0005 70 2.2242 (14) 2.9667 ( 7) 0.2301 (20) 0.0005 80 3.4558 (14) 4.5964 ( 7) 0.3472 (22) 0.0005 90 4.8076 (14) 6.4841 ( 7) 0.4257 (22) 0.0005 100 7.3914 (14) 9.6274 ( 7) 0.5927 (24) Table : Test 3 (3D). Performance achieved as the viscosity and mesh parameter vary. tol_outer= 10 9. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 20 / 23

Numerical experiments Test 4. Preconditioning strategy (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (yz(1 x 2 ), 0, e z ) Zero Dirichlet b.c. Other variable aggregations are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 21 / 23

Numerical experiments Test 4. Preconditioning strategy (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (yz(1 x 2 ), 0, e z ) Zero Dirichlet b.c. The discretization yields ɛt U + U(I ɛt + ɛt I ) + Υ 3 BU + Υ 1 U(Ψ 1 Φ 1 B) = 11 T where 1 is the vector of all ones Other variable aggregations are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 21 / 23

Numerical experiments Test 4. Preconditioning strategy (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (yz(1 x 2 ), 0, e z ) Zero Dirichlet b.c. The discretization yields ɛt U + U(I ɛt + ɛt I ) + Υ 3 BU + Υ 1 U(Ψ 1 Φ 1 B) = 11 T where 1 is the vector of all ones Υ 1 ῡ 1 I and obtain the preconditioning operator P : Y (ɛt + Υ 3 B)Y + Y(I ɛt + ɛt I + Ψ 1 ῡ 1 Φ 1 B) Other variable aggregations are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 21 / 23

Numerical experiments Test 4. Preconditioning strategy (3D) ɛ n x FGMRES+AGMG GMRES+MI20 FGMRES+KPIK 0.5000 60 1.2095 (15) 1.8236 ( 7) 1.2027 ( 6) 0.5000 70 2.1041 (15) 3.1649 ( 7) 1.6585 ( 6) 0.5000 80 3.7370 (16) 4.9765 ( 7) 2.4943 ( 6) 0.5000 90 7.5874 (16) 9.2040 ( 8) 3.2513 ( 6) 0.5000 100 7.7626 (16) 11.9912 ( 8) 4.7548 ( 6) 0.1000 60 2.1310 (18) - ( -) 1.5299 ( 8) 0.1000 70 2.8043 (18) - ( -) 1.8926 ( 8) 0.1000 80 5.1219 (19) - ( -) 3.2928 ( 9) 0.1000 90 7.3179 (19) - ( -) 4.6429 ( 9) 0.1000 100 9.5759 (19) - ( -) 6.5590 ( 9) 0.0500 60 1.4318 (18) - ( -) 1.7296 (10) 0.0500 70 2.8427 (19) - ( -) 2.5215 (10) 0.0500 80 4.9616 (20) - ( -) 3.6615 (10) 0.0500 90 7.1038 (20) - ( -) 5.0098 (10) 0.0500 100 10.7181 (21) - ( -) 6.8661 (10) Table : Test 4 (3D). Performance achieved as the viscosity and mesh parameter vary. - stands for excessive time in building the preconditioner. tol_outer= 10 9. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 22 / 23

Conclusions and outlook Conclusions and outlook Preliminary numerical experiments show that the new approach performs comparably well with respect to state-of-the-art approaches. Generalize the approach to more general settings overcoming the limitation to the use of uniform mesh on rectangle (parallelepipedal) domains. Modify the approach to handle dierent discretization strategies, e.g., SUPG (early attempts in [D.P., 2014, Master's thesis]). Reference: Matrix-equation-based strategies for convection-diusion equations D. Palitta and V. Simoncini ArXiv: 1501.02920 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 23 / 23

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