Svetlana Matculevich , Sergey Repin , and Pekka Neittaanm aki - - PowerPoint PPT Presentation

1/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

ON COMPUTABLE ESTIMATES OF THE DISTANCE TO THE EXACT SOLUTION OF PARABOLIC PROBLEMS BASED ON LOCAL POINCAR´ E TYPE INEQUALITIES

Svetlana Matculevich ∗†, Sergey Repin†∗, and Pekka Neittaanm¨ aki∗

∗Dept. of Mathematical Information Technology, University, of Jyv¨

askyl¨ a, Finland

†St. Petersburg Dept. of V.A. Steklov Institute of Mathematics of RAS, Russia

TIEJ601: Postgraduate Seminar in Information Technology

January 20, 2015

2/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

BACKGROUND

• Jun. 2010, BSc in Applied Mathematics and Informatics,

SPbSPU, Russia.

• Mar. 2012, MSc in Information Technology, University of

Jyv¨ askyl¨ a, Finland.

• Jun. 2012, MSc in Mathematical Modeling and Informatics,

SPbSPU, Russia.

3/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

PHD TERMS AND DEFENSE

PhD started on June 1, 2012. The defense is planned in Autumn, 2015. The thesis structure is the collection of papers.

4/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

SUPERVISION

• Prof. Sergey Repin

St.Petersburg Department of Steklov Mathematical Institute RAS, Russia MIT, University of Jyv¨ askyl¨ a, Finland

• Prof. Pekka Neittaanm¨

aki MIT, University of Jyv¨ askyl¨ a, Finland

5/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

THE PROGRESS

69 ECTs completed 1 + 3 journal paper accepted:

• S. Matculevich and S.Repin, Computable bounds of the

distance to the exact solution of parabolic problems based on Poincar´ e type inequalities, Zap. Nauchn. Sem. S.-Peterburg Otdel Mat. Inst. Steklov (POMI), 425(1), 7–34, 2014.

• S. Matculevich, P.Neitaanm¨

aki, and S.Repin, A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne–Weinberger inequality, Discrete and Continuous Dynamical Systems - Series A, AIMS, 35(6), 2659–2677, 2015.

• S. Matculevich and S.Repin, Computable estimates of the

distance to the exact solution of the evolutionary reaction-diffusion equation, Applied Mathematics and Computation, Elsevier, 247, 329–347, 2014.

6/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

STAGES OF MATHEMATICAL MODELING

1

Creating mathematical model of a certain phenomenon := described by the system of partial differential equation (PDE’s).

2

Solving the system := constructing numerical representation of the key characteristics of phenomena.

3

Analyze the results := comparison of the numerical data with physical properties of the object.

4

Conclusion.

7/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

STAGE OF THE reliable MATHEMATICAL EXPERIMENT

Mathematical model is correct. ⇒ Existence theory of PDEs: existence and stability of exact solution. Numerical method is correct. ⇒ Approximation theory of PDEs: convergence, stability, and ect. Errors (of approximation or numerical method) are explicitly controlled. ⇒ A posteriori error control theory for PDEs.

8/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPLICATIONS OF EVOLUTIONARY REACTION-DIFFUSION EQUATIONS

Heat conduction (transfer) in thermodynamics:

modeling thermal energy storage, heat transfer in human body.

8/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPLICATIONS OF EVOLUTIONARY REACTION-DIFFUSION EQUATIONS

Heat conduction (transfer) in thermodynamics:

modeling thermal energy storage, heat transfer in human body.

Diffusion-convection-reaction in chemistry and biology:

population dynamics, e.i. predator-prey system, ecological dynamics.

9/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

MOTIVATION

To control of the quality of numerical computation for

EVOLUTIONARY CLASS OF PROBLEMS

∂u ∂t = Au + f, t ∈ (0, T), u(0) = u0, (1)

9/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

MOTIVATION

To control of the quality of numerical computation for

EVOLUTIONARY CLASS OF PROBLEMS

∂u ∂t = Au + f, t ∈ (0, T), u(0) = u0, (1) by equipping numerical methods with error estimates M(v, D) ≤| | | u − v | | | ≤ M(v, D), which must be (2) explicitly computable, universal, reliable, efficient.

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary,

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary, QT := Ω × (0, T), T > 0, and

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary, QT := Ω × (0, T), T > 0, and ST := ∂Ω × [0, T] = ΓD × [0, T] = SD, where ΓD is part of boundary with Dirichlet BC.

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary, QT := Ω × (0, T), T > 0, and ST := ∂Ω × [0, T] = ΓD × [0, T] = SD, where ΓD is part of boundary with Dirichlet BC. Find u(x, t) ∈ H1

0(QT) satisfying the following system:

ut − ∇ · A∇u + λu = f ∈ L2(QT) in QT, u(x, 0) = ϕ ∈ L2(Ω) in Ω, u = 0

• n SD,

where H1

0(Ω) subspace of H1(Ω) with functions satisfying the Dirichlet

boundary condition.

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary, QT := Ω × (0, T), T > 0, and ST := ∂Ω × [0, T] = ΓD × [0, T] = SD, where ΓD is part of boundary with Dirichlet BC. Find u(x, t) ∈ H1

0(QT) satisfying the following system:

ut − ∇ · A∇u + λu = f ∈ L2(QT) in QT, u(x, 0) = ϕ ∈ L2(Ω) in Ω, u = 0

• n SD,

where H1

0(Ω) subspace of H1(Ω) with functions satisfying the Dirichlet

boundary condition.

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary, QT := Ω × (0, T), T > 0, and ST := ∂Ω × [0, T] = ΓD × [0, T] = SD, where ΓD is part of boundary with Dirichlet BC. Find u(x, t) ∈ H1

0(QT) satisfying the following system:

ut − ∇ · A∇u + λu = f ∈ L2(QT) in QT, u(x, 0) = ϕ ∈ L2(Ω) in Ω, u = 0

• n SD,

where H1

0(Ω) subspace of H1(Ω) with functions satisfying the Dirichlet

boundary condition.

10/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary, QT := Ω × (0, T), T > 0, and ST := ∂Ω × [0, T] = ΓD × [0, T] = SD, where ΓD is part of boundary with Dirichlet BC. Find u(x, t) ∈ H1

0(QT) satisfying the following system:

ut − ∇ · A∇u + λu = f ∈ L2(QT) in QT, u(x, 0) = ϕ ∈ L2(Ω) in Ω, u = 0

• n SD,

where H1

0(Ω) subspace of H1(Ω) with functions satisfying the Dirichlet

boundary condition.

11/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

WEAK FORMULATION OF THE PROBLEM

Find u(x, t) ∈ H1

0(QT) [4, 5, 6, 15] satisfying the integral identity

• Ω
• u(x, T)η(x, T) − u(x, 0)η(x, 0)
• dx −
• QT

uηt dxdt+

• QT

A∇u · ∇η dxdt +

• QT

λuη dxdt =

• QT

fη dxdt, ∀η ∈ H1

0(QT).

(3)

[4] S. I. Repin, Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation, Rend. Mat. Acc. Lincei, 2002. [8] S. Repin and S. Sauter, Functional a posteriori estimates for the reaction-diffusion problem, C. R. Acad. Sci. Paris, 2006.

12/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

ERROR ENERGY NORM

Let u, v ∈ H1

0(Ω), then

e(x, t) = (u − v)(x, t) is measured by the norm [e](ν,θ,ζ) = ν | | |∇e| | |2

A + θ w(θ, χ, λ)e 2 QT + ζe (x, T)2 Ω,

(4) where | | |∇e| | |2

A :=

• QT

A∇e · ∇e dx dt, ν, θ, χ, and ζ are positive numbers, and w is positive function.

13/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

MAIN IDEA TO OBTAIN THE UPPER BOUND

To introduce the auxiliary vector-function y ∈ Ydiv (QT), where Ydiv (QT) :=

• y ∈ L2(Ω, Rd) | div y ∈ L2(Ω), for a.e. t ∈ (0, T)
• ,

(5) which satisfies the relation

• QT

div y η dxdt +

• QT

y · ∇η dxdt = 0, for a.e. t ∈ (0, T). (6)

13/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

MAIN IDEA TO OBTAIN THE UPPER BOUND

To introduce the auxiliary vector-function y ∈ Ydiv (QT), where Ydiv (QT) :=

• y ∈ L2(Ω, Rd) | div y ∈ L2(Ω), for a.e. t ∈ (0, T)
• ,

(5) which satisfies the relation

• QT

div y η dxdt +

• QT

y · ∇η dxdt = 0, for a.e. t ∈ (0, T). (6)

[9] P. Neittaanm¨ aki and S. Repin, Reliable methods for computer simulation. Error control and a posteriori estimates. Studies in Mathematics and its Applications, 33, Elsevier Science B.V., Amsterdam, 2004. [12] S. Repin, A posteriori error estimates for partial differential equations. Radon Series on Computational and Applied Mathematics, Walter de Gruyter, Berlin, 2008. [7] O. Mali, P. Neittaanm¨ aki, and S. Repin, Accuracy verification methods. Theory and

• algorithms. Springer, 2014.

14/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

TWO-SIDED ERROR BOUND

Let v ∈ H1

0(QT),

14/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

TWO-SIDED ERROR BOUND

Let v ∈ H1

0(QT),

y ∈ Ydiv (QT) (which approximate ∇u),

14/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

TWO-SIDED ERROR BOUND

Let v ∈ H1

0(QT),

y ∈ Ydiv (QT) (which approximate ∇u), η ∈ H1

0(QT)

(which mimics u − v),

14/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

TWO-SIDED ERROR BOUND

Let v ∈ H1

0(QT),

y ∈ Ydiv (QT) (which approximate ∇u), η ∈ H1

0(QT)

(which mimics u − v), M2(η, v) ≤ [u − v](ν,θ,ζ) ≤ M

2(v, y),

(7) where ν, θ, χ, and ζ are positive numbers, and w is positive function.

14/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

TWO-SIDED ERROR BOUND

Let v ∈ H1

0(QT),

y ∈ Ydiv (QT) (which approximate ∇u), η ∈ H1

0(QT)

(which mimics u − v), M2(η, v) ≤ [u − v](ν,θ,ζ) ≤ M

2(v, y),

(7) where ν, θ, χ, and ζ are positive numbers, and w is positive function.

15/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

Theorem (Error majorant)

Let v ∈ ˚ H1(QT), y ∈ Ydiv (QT), δ ∈ (0, 2], γ(t) ≥ 1, µ(x) is a real-valued function taking values in [0, 1], and

1 α1(t) + 1 α2(t) = δ. Then,

(2 − δ)| | |∇e| | |2

QT +

• 2 − 1

γ

• √

λe

• 2

QT

+ e(x, T)2

Ω =: [e](2−δ, 2− 1

γ , 1)

≤ M

2(v, y) := e(0, x)2 Ω + T

• γ
• µ

√ λ Rf(v, y)

• 2

Ω

+ α1(t, δ)C2

FΩ

ν1 (1 − µ) Rf(v, y)2

Ω + α2(t, δ)Rd(v, y)2 A−1

• dt,

(8) where CFΩ is Friedrichs constant, and Rf(v, y) = f − vt − λv + div y, Rd(v, y) = y − A∇v. (9)

16/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

Theorem (Error majorant)

Let v ∈ ˚ H1(QT), y ∈ Ydiv (QT), φ(x) = v(x, 0), δ = 1, λ(x, t) = 0, µ(x, t) = 0, and α1 = 1

δ

• 1 + 1

β

• , α2 = 1

δ

• 1 + β
• , β = const. Then,

| | |∇(u − v)| | |2

QT + (u − v)(x, T)2 Ω =: [u − v](2−δ,0,1) ≤

M

2(v, y, β) := T

• 1 + β
• m2

d +

• 1 + 1

β

• C2

FΩmf I

• dt,

(10) where CFΩ is Friedrichs constant, and m2

d = y − ∇v2 Ω

and mf

I = f − vt + div y2 Ω.

(11)

[16] S. Matculevich and S. Repin, Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation. Appl. Math. and Comput., 247:329–347, 2014. [17] S. Matculevich, P. Neittaanm¨ aki, and S. Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne–Weinberger

• inequality. Disc. and Cont. Dyn. Sys. - A, 35(6), 2015.

17/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

DESCRITIZATION OF QT

TK =

K−1

• k=0

[tk, tk+1] is mesh selected on [0, T], QT =

K−1

• k=0

Qk, where Qk := (tk, tk+1) × Ω. t0 t1 Q0 ... Qk−1 Qk Qk+1 ϕ(x) vk

τh

vk+1

τh

vk+2

τh

tk−1 tk tk+1 tk+2

Figure: Discretization of the time-cylinder QT.

18/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EXAMPLE 1. UNIT DOMAIN IN R2 (FENICS)

(0, 1) × (0, 1) ∈ R2, T = 1, A = I, ϕ = 0 on SD, f = x (1 − x) y (1 − y) (2t + 1) + 2

• t2 + t + 1
• (x (1 − x) + y (1 − y)) in

QT The exact solution is u = x (1 − x) y (1 − y)

• t2 + t + 1
• .

18/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EXAMPLE 1. UNIT DOMAIN IN R2 (FENICS)

(0, 1) × (0, 1) ∈ R2, T = 1, A = I, ϕ = 0 on SD, f = x (1 − x) y (1 − y) (2t + 1) + 2

• t2 + t + 1
• (x (1 − x) + y (1 − y)) in

QT The exact solution is u = x (1 − x) y (1 − y)

• t2 + t + 1
• .

Initial mesh is Θ K×N1×N2 = Θ 20×5×5. Marking criterion is refinement of 20 % of elements with the highest error. On Qk = [tk, tk+1] × Ω:

v = vk(x) tk+1−t

τ

+ vk+1(x) t−tk

τ ,

y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ , τ = tk+1 − tk on Qk T.

vk(x) ∈ P1, yk(x) ∈ RT2.

19/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

INTEGRAL ERROR ESTIMATION

tk [e] 2

(1, 0, 1)

M IM

eff := M [e](ν, θ, ζ) ≥ 1

0.05 1.34e-04 1.54e-04 1.07 0.15 2.89e-04 3.52e-04 1.10 0.25 3.95e-04 4.81e-04 1.10 0.35 4.79e-04 5.80e-04 1.10 0.45 5.38e-04 6.49e-04 1.10 0.55 5.89e-04 7.05e-04 1.09 0.65 6.32e-04 7.53e-04 1.09 0.75 6.64e-04 7.86e-04 1.09 0.85 6.91e-04 8.15e-04 1.09 0.95 7.15e-04 8.40e-04 1.08

Table: Example 1. The integral error, majorant, and its efficiency index.

20/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

RESULT MESH FOR THE REFINEMENT OBTAIN BASED

ON TIME LEVEL k = 19

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

(a) error distr., DOFs = 1829

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

(b) indic. distr., DOFs = 1845 Figure: Example 1. Result mesh for the refinement obtain based on error distribution and majorant distribution (marking strategy is predefined amount of elements to refine 20 %).

21/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

L-SHAPE DOMAIN IN R2 (FENICS)

Ω = ([−1, 1] × [0, 1]) ∪ ([−1, 0] × [−1, 0]) ∈ R2, T = 1, A = I, ϕ = (x2 + y2)1/3 sin 2

3atan(y, x)

• n SD,

f = (x2 + y2)1/3 sin 2

3atan(y, x)

• (2 t + 1) in QT.

The exact solution is u = (x2 + y2)1/3 sin 2

3atan(y, x)

t2 + t + 1

• .

vk(x) ∈ P1, yk(x) ∈ RT2; Marking criterion is refinement of 20 % of elements with the highest error. initial mesh:

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

Figure: Initial mesh on l-shape domain DOFs = 35.

22/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

MESHES OBTAINED ON TIME LEVEL k = 19

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

(a) error distr., DOFs = 2061

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

(b) indic. distr., DOFs = 1887 Figure: Result mesh for the refinement obtain based on error distribution and majorant distribution (marking strategy is predefined amount of elements to refine 20 %).

23/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

EVOLUTIONARY REACTION-DIFFUSION PROBLEM WITH mixed BC AND Ω of complicated shape

Ω ∈ Rd is a bounded connected domain with Lipschitz boundary ∂Ω = ΓD ∪ ΓN, QT := Ω × (0, T), T > 0, and ST := ∂Ω × [0, T] = (ΓD ∪ ΓN) × [0, T] = SD ∪ SN. Find u(x, t) ∈ H1

0(QT) satisfying the following system:

ut − ∇ · A∇u + λu = f ∈ L2(QT) in QT, u(x, 0) = ϕ ∈ L2(Ω) in Ω, u = 0

• n SD,

A∇u · n = g ∈ L2(SN)

• n SN.

(12)

24/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

Theorem (Error majorant)

Let v ∈ ˚ H1(QT), y ∈ Ydiv (QT), δ ∈ (0, 2], γ(t) ≥ 1, µ(x) is a real-valued function taking values in ∈ [0, 1], and

1 α1(t,δ) + 1 α2(t,δ) + 1 α3(t,δ) = δ. Then,

(2− δ)| | |∇e| | |2

QT +

• 2− 1

γ

• √

λe

• 2

QT

+ e(x, T)2

Ω =: [e](2−δ, 2− 1

γ , 1)≤M

2(v, y)

:= e(0, x)2

Ω + T

• γ
• µ

√ λ Rf(v, y)

• 2

Ω

+ α1(t, δ)C2

FΩ

ν1 (1 − µ) Rf(v, y)2

Ω

+ α2(t, δ)Rd(v, y)2

A−1 + α3(t, δ)C2 tr

ν1

• g − y · n
• 2

ΓN

• dt,

(13) where contains the constants CFΩ and Ctr in the Friedrichs and trace type inequalities wΩ ≤ CFΩ ∇w Ω, (14) wΓN ≤ Ctr ∇w Ω , (15) which are valid for functions in H1

0(Ω).

25/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

DECOMPOSITION OF DOMAIN Ω

1) Ω :=

• Ωi⊂ OΩ

Ωi, Ωi ∩ Ωj = ∅, i = j, i, j = 1, . . . , N, 2) Γij = Ωi ∩ Ωj, ΓDk = Ωk ∩ ΓD, ΓNi = Ωi ∩ ΓN. Ωi Ωj Ωk ΩN ΓD ΓN Γij ΓDk ΓNi

26/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

POINCAR´

E INEQUALITIES AND CORRESPONDING CONSTANTS

Subdomains Ωi are convex domains with Lipschitz boundaries.

• H1(Ωi) :=
• w ∈ H1(Ωi)
• w
• Ωi = 0
• , where {w}Ωi :=

1 |Ωi|

• Ωi

w dx. ∀w ∈ H1(Ωi) the classical Poincar´ e inequality [11]: wΩi ≤ CPΩi∇wΩi. (16)

26/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

POINCAR´

E INEQUALITIES AND CORRESPONDING CONSTANTS

Subdomains Ωi are convex domains with Lipschitz boundaries.

• H1(Ωi) :=
• w ∈ H1(Ωi)
• w
• Ωi = 0
• , where {w}Ωi :=

1 |Ωi|

• Ωi

w dx. ∀w ∈ H1(Ωi) the classical Poincar´ e inequality [11]: wΩi ≤ CPΩi∇wΩi. (16) Due to result of Payne, Weinberger 1960 [10] with corrections of Bebendorf 2013 [2], we have CPΩi ≤ diam Ωi π .

27/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

POINCAR´

E TYPE INEQUALITIES AND CORRESPONDING CONSTANTS

Consider Ωi with the edge(face) T is a part of the boundary ∂Ωi, which coincides with Γij or ΓNi. Let H1(Ωi, T ) :=

• w ∈ H1(Ωi)
• w
• T = 0
• ,

then, ∀w ∈ H1(Ωi, T ) we have wT ≤ CT Ωi∇wΩi. (17)

27/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

POINCAR´

E TYPE INEQUALITIES AND CORRESPONDING CONSTANTS

Consider Ωi with the edge(face) T is a part of the boundary ∂Ωi, which coincides with Γij or ΓNi. Let H1(Ωi, T ) :=

• w ∈ H1(Ωi)
• w
• T = 0
• ,

then, ∀w ∈ H1(Ωi, T ) we have wT ≤ CT Ωi∇wΩi. (17) Due to [8], CT Ωi are found for right quadrilateral in R2 and R3, where T is one of the sides, right and isosceles triangles in R2, where T is the leg or hypotenuse.

[8] A. I. Nazarov and S. I. Repin, Exact constants in Poincare type inequalities for functions with zero mean boundary traces. ArXiv, math/1211.2224, 2012.

28/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

CONDITION 1

• (1 − µ)(f + div y − vt − λv)
• Ωk⊂ O0

= 0, for a.a. t ∈ [0, T], (18) where Ω0 :=

• Ωk⊂ O0

Ωk, O0 :=

• Ωk ⊂ OΩ
• λ|Ωk < P, k = 1, . . . , N0
• .

ΩP Ωi Ωj Ωk ΓD ΓN Γij ΓDk ΓNi

29/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

CONDITION 2

• g − y · n
• ΓNj⊂SN

= 0, for a.a. t ∈ [0, T], (19) where ΓNj = ∂Ωj ∩ ΓN, j = 1, . . . , M, M ≤ N, and SN denotes a collection of non-overlapping faces ΓNj. ΩP Ωi Ωj Ωk ΓD ΓN Γij ΓDk ΓNi ΓNj1 ΓNj2 Γ

30/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

Theorem (Majorant based on local Poincar´ e type inequalities)

Assume that Conditions 1 and 2 hold, then for any v ∈ H1

0(QT) and y ∈ Ydiv (QT),

δ ∈ (0, 2], ρ1(t) ≥ 1, ρ2(t) ≥ 1, µ(x) ∈ [0, 1], λ(x) > 0, α1(t), α2(t), α3(t) are positive scalar-valued functions satisfying the relation

1 α1(t) + 1 α2(t) + 1 α3(t) = δ, we have the

estimate [e] 2

(ν, θ, 1, 2) ≤ M 2 I,N(v, y; δ, ρ1, ρ2, µ):= T

• ρ1
• 1

√ λ rf, µ(v, y)

• 2

Ω

+ ρ2

• Ωl⊂ OP

|Ωl| P2

• rf,1−µ(v, y)

2

Ωl

+ α1(t)rA(v, y) 2

A−1,Ω

+ α2(t)

• Ωl⊂ OP

CPΩ2

l

λA

• rf,1−µ(v, y)
• 2

Ωl +

• Ωk⊂ O0

CPΩ2

k

λA

• rf,1−µ(v, y)
• 2

Ωk

• + α3(t)
• ΓNj⊂ SN

CΓΩ2

j

λA

rF(v, y)2

ΓNj

• dt,

ν = 2 − δ, θ(x) =

• λ(x)
• 2 −

1 ρ1(t) − 1 ρ2(t)

1

2 are positive weights.

31/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

REFERENCE CASE WITH TRIANGLE Tleg ∈ R2

x1 x2 (0, 0) (1, 0) (0, 1)

• Γ
• Tleg

y1 y2 (0, 0) (h, 0)

• hρ cos(α), hρ sin(α)
• Γ

α T Fleg

Figure: Mapping Fleg : ˆ Tleg → T.

with y = Fleg (x), = Blegx where Bleg =

• h

ρh cos(α) ρh sin(α)

• ,

J(Fleg) = ρh2 sin(α), (20)

32/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

REFERENCE CASE WITH TRIANGLE Thyp ∈ R2

x1 x2 (0, 0) (1, 0) ( 1

2, 1 2)

• Γ
• Thyp

y1 y2 (0, 0) (h, 0) (hρ cos(α), hρ sin(α)) Γ α T Fhyp

Figure: Mapping Fhyp : Tleg → T.

with Fhyp(x) = Bhyp x, where Bhyp =

• h

2ρh cos(α) − h 2ρh sin(α)

• ,

J(Fhyp) = 2ρh2 sin(α). (21)

33/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

POINCAR´

E TYPE INEQUALITIES ON ARBITRARY T ∈ R2

∀v ∈ H1(T, Γ) on T = conv

• (0, 0), (h, 0),
• hρ cos(α), hρ sin(α)
• , where

ρ > 0, with Γ :=

• x1 ∈ [0, h]; x2 = 0
• , the Poincar´

e type inequalities read as vT ≤ CPT h ∇vT, CPT = min

• c+

p,leg CPˆ Tleg, c+ p,hyp CPˆ Thyp

• ,

(22) vΓ ≤ CΓT h

1 2 ∇vT,

CΓT = min

• c+

γ,leg CΓˆ Tleg, c+ γ,hyp CΓˆ Thyp

• ,

(23) where c+

p,leg = µleg 1 2 , c+ γ,leg =

• µleg

ρ sin α

1

2 , c+ p,hyp = µhyp 1 2 , c+ γ,hyp =

• µhyp

2ρ sin α

1

2 ,

(24) and constants µhyp and µleg are defined as µleg(ρ, α) = 1

2

• 1 + ρ2 +
• 1 + ρ4 + 2 cos(2α) ρ2 1

2

• ,

and µhyp(ρ, α) = 2ρ2 − 2ρ cos(α) + 1 +

• (2ρ2 + 1)(2ρ2 + 1 − 4ρ cos(α) + 4ρ2 cos(2α))

1

2 ,

respectively.

34/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPROXIMATION OF THE POINCAR´

E CONTANTS IN R2

Consider w defined in space H1(T, Γ), which is equipped with basis Φ :=

• ϕij = xiyj
• r cos(πix) cos(πjy)),

i, j = 0, . . . , N

• .

(25) Minimize the generalized Rayleigh quotients R PT(S, K; ξ) := ξTSξ

ξTKξ ≈ ∇vT v−{v}TT

and R ΓT(S, L; ξ):= ξTSξ

ξTLξ ≈ ∇vT v−{v}ΓΓ ,

where ξ is non-zero vector, and matrices K, L, and S are generated as follows: Lk,m =

• Γ

φkφm ds, Kk,m =

• T

φkφm dx, Sk,m =

• T

∇φk · ∇φm dx, k, m = 1, . . . , M.

35/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPROXIMATION OF THE POICAR´

E CONSTANTS R2

Power series Fourier series N M

• CP

T

• CΓ

T

• CP

T

• CΓ

T

1 3 0.3820 0.6268 0.3794 0.6090 2 8 0.4045 0.7120 0.3934 0.6618 3 15 0.4074 0.7136 0.3977 0.6799 4 24 0.4075 0.7136 0.4001 0.6891 5 35 0.4075 0.7136 0.4016 0.6945 6 48 0.4075 0.7136 0.4026 0.6981

Table: Convergence of the constants for Tleg (α = π

2 , ρ = 1) with

respect to the size of basis M generated by power and Fourier series.

36/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPROXIMATION CPT WITH UPPER BOUND C+

PT

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 pi 0.3 0.4 0.5 0.6 0.7 α C+

Pˆ Tleg

C+

Pˆ Thyp

C+

PT

• CPT

(a) ρ =

√ 2 2

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 pi 0.3 0.4 0.5 0.6 0.7 0.8 α C+

Pˆ Tleg

C+

Pˆ Thyp

C+

PT

• CPT

(b) ρ = 1

37/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPROXIMATION CΓT WITH UPPER BOUND CΓT

+

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 pi 1 2 3 4 5 α C+

Γˆ Tleg

C+

Γˆ Thyp

C+

ΓT

• CPT

(a) ρ =

√ 2 2

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 pi 1 2 3 4 5 α C+

Γˆ Tleg

C+

Γˆ Thyp

C+

ΓT

• CPT

(b) ρ = 1

√

38/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

REFERENCE CASE WITH TRIANGLE Tπ/2,ρ ∈ R2

• A
• B
• C
• D

x y z

• Tπ/2,ρ
• Γ

F A B C D T

• Γ

x y z

Figure: Tetrahedron T ∈ R3, where

• Tπ/2,ρ =
• (0, 0, 0), (1, 0, 0), (0, 0, 1), (0, ρ, 0)
• and

T =

• (0, 0, 0), (h, 0, 0), (0, 0, H), (hρ cos(α), hρ sin(α), 0)
• .

39/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPROXIMATE CPT AND CΓT WITH ESTIMATES, ρ = 1.

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 α C+

Pˆ Tπ/2

C+

Pˆ Tπ/3

C+

PT

• CPT

(a)

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 0.8 1 1.2 1.4 1.6 1.8 2 α C+

Γˆ Tπ/2

C+

Γˆ Tπ/3

C+

ΓT

• CΓT

(b)

40/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

APPROXIMATE CPT AND CΓT WITH ESTIMATES, ρ =

√ 2 2 .

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 0.3 0.4 0.5 0.6 0.7 0.8 α C+

Pˆ Tπ/2

C+

Pˆ Tπ/4

C+

PT

• CPT

(c)

pi/6 pi/4 pi/3 pi/2 2*pi/3 3*pi/4 5*pi/6 1 1.5 2 2.5 3 α C+

Γˆ Tπ/2

C+

Γˆ Tπ/4

C+

ΓT

• CΓT

(d)

41/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

SUMMARY

RESULTS:

1

We have obtained error estimates, which are applicable for real problems with domain of complicated shape and non-trivial BC.

2

Instead of global constants CFΩ and CΓΩ, which are

hard to compute and much worse value-wise (since they are related to the whole Ω),

we use local Poincar´ e constants, which

we know exactly or can provide guaranteed estimate, and are sharper (since they are related to the local subdomains Ωi).

TO DO:

1

Provide the numerical tests justifying theoretical results.

42/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

THANK YOU FOR YOUR ATTENTION!

42/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

• G. Acosta and R. G. Dur´

an. An optimal Poincar´ e inequality in L1 for convex domains.

• Adv. Numer. Anal., pages Art. ID 164519, 15, 2009.
• M. Bebendorf.

A note on the Poincar´ e inequality for convex domains.

• Z. Anal. Anwendungen, 22(4):751–756, (2003).

S.-K. Chua and R. L. Wheeden. Estimates of best constants for weighted Poincar´ e inequalities on convex domains.

• Proc. London Math. Soc. (3), 93(1):197–226, 2006.
• L. C. Evans.

Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.

• O. A. Ladyzhenskaya.

The boundary value problems of mathematical physics. Springer, New York, 1985.

• O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva.

42/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

Linear and quasilinear equations of parabolic type. Nauka, Moscow, 1967.

• O. Mali, P. Neittaanm¨

aki, and S. I. Repin. Accuracy verification methods. Theory and algorithms (in print). Springer, 2014.

• A. I. Nazarov and S. I. Repin.

Exact constants in Poincare type inequalities for functions with zero mean boundary traces. ArXiv, math/1211.2224, 2012.

• P. Neittaanm¨

aki and S. I. Repin. Reliable methods for computer simulation. Error control and a posteriori estimates, Volume 33 of Studies in Mathematics and its Applications. Elsevier Science B.V., Amsterdam, 2004.

• L. E. Payne and H. F. Weinberger.

An optimal Poincar´ e inequality for convex domains.

• Arch. Rational Mech. Anal., 5:286–292 (1960), 1960.
• H. Poincare.

Sur les Equations aux Derivees Partielles de la Physique Mathematique.

• Amer. J. Math., 12(3):211–294, 1890.

42/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

• S. Repin.

A posteriori estimates for partial differential equations, Walter de Gruyter, Berlin, 2008.

• S. Repin and S. Sauter.

Functional a posteriori estimates for the reaction-diffusion problem.

• C. R. Acad. Sci. Paris, 343(1):349–354, 2006.
• S. Repin.

Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation.

• Rend. Mat. Acc. Lincei, 13(9):121–133, 2002.
• J. Wloka.

Partial Differential Equations. Cambridge University Press, 1987.

• S. Matculevich and S. Repin.

Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation.

• Appl. Math. and Comput., 247:329–347, 2014.
• S. Matculevich, P. Neittaanm¨

aki, and S. Repin.

42/42 INTRODUCTION GENERAL ERROR ESTIMATES EXAMPLES APPLICATION FOR ADVANCED PROBLEMS ADVANCED ERROR ESTIMATE

A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne–Weinberger inequality. Discrete and Continuous Dynamical Systems - Series A, 35(6), 2015.

• S. Matculevich and S. Repin.

Computable bounds of the distance to the exact solution of parabolic problems based on Poincar´ e type inequalities.

• Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), 425:7–34,

2014.