On the global convergence of a singularly perturbed parabolic - - PowerPoint PPT Presentation

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On the global convergence of a singularly perturbed parabolic - - PowerPoint PPT Presentation

Oct 07, 2023 584 likes •974 views

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh On the global convergence of a singularly perturbed parabolic problem of reaction diffusion type with a discontinuous initial condition J.L. Gracia, E.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

On the global convergence of a singularly perturbed parabolic problem of reaction diffusion type with a discontinuous initial condition

J.L. Gracia, E. O’Riordan

Department of Applied Mathematics School of Mathematics University of Zaragoza (Spain) Dublin City University (Ireland)

Workshop “Numerical Analysis for singularly perturbed Problems” Dedicated to the 60th birthday of Martin Stynes 16th–18th November 2011, TU Dresden Research supported by the project MEC/FEDER MTM 2010-16917 and the Diputaci´

  • n General de Arag´
  • n
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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Martin Happy 60th!!!!

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

On the global convergence of a singularly perturbed parabolic problem of reaction diffusion type with a discontinuous initial condition

J.L. Gracia, E. O’Riordan

Department of Applied Mathematics School of Mathematics University of Zaragoza (Spain) Dublin City University (Ireland)

Workshop “Numerical Analysis for singularly perturbed Problems” Dedicated to the 60th birthday of Martin Stynes 16th–18th November 2011, TU Dresden Research supported by the project MEC/FEDER MTM 2010-16917 and the Diputaci´

  • n General de Arag´
  • n
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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Outline

Talk is split into two parts.

  • 1. Fitted mesh method for reaction-diffusion problem with smooth

initial condition, but containing a layer in the initial condition.

  • 2. Reaction-diffusion problem with discontinuous data.

1 Fitted operator method (with uniform mesh) when initial

condition is discontinuous

2 Approximate problem by regularizing initial condition 3 Fitted mesh method (with classical operator) 4 Fitted operator combined with fitted mesh method

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Uniformly convergent schemes

We are interesting in robust schemes (or uniformly convergent schemes) max

0<ε≤1 UN ε − uε∞ ≤ CN−p,

p > 0 where C is a constant independent of the discretization parameters N and (also) the singular perturbation parameters ε. A priori numerical methods: Fitted operator methods (Il’in, El-Mistikawya and Werle, ...) (Quasi–)Uniform mesh + special discrete operator Fitted mesh methods Layer–adapted meshes (Bakhvalov, Shishkin, ...) + classical discrete operator

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Parabolic reaction-diffusion problem I

Lεu := −εuxx + b(x, t)u + p(x, t)ut = f (x, t), (x, t) ∈ (0, 1) × (0, T], u(x, 0) = φ(x; ε, d), 0 ≤ x ≤ 1, u(0, t) = 0, u(1, t) = 1, 0 < t ≤ T, b(x, t) ≥ β > 0, p(x, t) ≥ p0 > 0, where the initial condition φ is smooth, but is constructed to have an artificial interior layer in the vicinity of a point x = d, 0 < d < 1 and C√ε ≤ d ≤ 1 − δ < 1. φ(x; ε, d) =    0, if x ∈ Ω− := (0, d), K

  • 1 − e−
  • β1

ε (x−d)p

, if x ∈ Ω+ := (d, 1), where p ≥ 4, β1 ≥ 2β > 0, K are such that φ(1; ε, d) = 1. Throughout Q− := Ω− × (0, T], Q+ := Ω+ × (0, T].

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Solution decomposition

Maximum Principle The differential operator satisfies a maximum principle, and from this we deduce that u ¯

Q ≤ C,

where u ¯

Q = max(x,t)∈ ¯ Q |u(x, t)| is the maximum norm.

To identify the layer structure of the solution, we consider a decomposition of the solution of the form u = v + w + z, into a discontinuous regular component v, a continuous boundary layer component w and a discontinuous interior layer component z.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Theorem There exists functions r0(t), r1(t), r2(t) such that the solutions v−, v+ of the problems Lεv− = f , (x, t) ∈ Q−, v−(0, t) = 0, v−(x, 0) = 0, Lεv+ = f , (x, t) ∈ Q+, v+(x, 0) = 1, v−(d, t) = r0(t), v+(d, t) = r1(t), v+(1, t) = r2(t), satisfy v− ∈ C4+γ( ¯ Q−), v+ ∈ C4+γ( ¯ Q+) and for 0 ≤ j + 2m ≤ 4 the bounds

  • ∂j+mv−

∂xj∂tm

  • ¯

Q−

≤ C(1 + ε1−j/2),

  • ∂j+mv+

∂xj∂tm

  • ¯

Q+

≤ C(1 + ε1−j/2).

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Decompose u − v into two subcomponents u − v = w + z The boundary layer function w is the solution of w ≡ 0, (x, t) ∈ ¯ Q−; Lεw = 0, (x, t) ∈ Q+; w(d, t) = 0, w(x, 0) ≈ 0, w(1, t) = u(1, t) − v+(1, t), Theorem The boundary layer function w ∈ C0( ¯ Q) ∩ C4+γ( ¯ Q+) and

  • ∂j+mw

∂xj∂tm (x, t)

  • ≤ Cε−j/2e−
  • β

ε (1−x), 0 ≤ j +2m ≤ 4, (x, t) ∈ Q+.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

The discontinuous multi-valued interior layer function z satisfies Lεz = 0, (x, t) ∈ Q− ∪ Q+, [z](d, t) = −[v](d, t), [zx](d, t) = −[vx + wx](d, t), t ≥ 0, z(0, t) = z(1, t) = 0, t ≥ 0, z(x, 0) = 0, x ∈ Ω−, z(x, 0) ≈ e−

  • β

ε (x−d), x ∈ Ω+.

Theorem Assume C√ε ≤ d ≤ 1 − δ < 1. The interior layer function z ∈ C4+γ( ¯ Q−) ∪ C4+γ( ¯ Q+) and for 0 ≤ j + 2m ≤ 4

  • ∂j+mz

∂xj∂tm (x, t)

  • ≤

Cε−j/2e−

  • β

ε (d−x), (x, t) ∈ Q−,

  • ∂j+mz

∂xj∂tm (x, t)

  • ≤

Cε−j/2e−

  • β

ε (x−d), (x, t) ∈ Q+.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Numerical method

−εδ2

xUN,M + b(xi, tj)UN,M + p(xi, tj)D− t UN,M = f (xi, tj), tj > 0,

UN,M(xi, 0) = φ(xi; ε, d), 0 < xi < 1, UN,M(0, tj) = 0, UN,M(1, tj) = 1, tj ≥ 0. Split the space domain using a Shishkin mesh [0, d − τ1] ∪ [d − τ1, d + τ2] ∪ [d + τ2, 1 − τ2] ∪ [1 − τ2, 1], with τ1 = min{d 2 , 2 ε β ln N}, τ2 = min{1 − d 4 , 2 ε β ln N}.

INTERIOR LAYER BOUNDARY LAYER

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Theorem (G., O’Riordan, submitted) Let u be the solution of the continuous problem and let ¯ UN be the linear interpolant of the solution of the discrete problem. Then, ¯ UN,M − u ¯

Q ≤ C((N−1 ln N)2 + N−1ε1/2 + ∆t).

Limiting case of d = 0 Consider the following initial condition φ(x; ε) = K

  • 1 − e−
  • β1

ε xp

, p ≥ 4. The space domain split into [0, σc] ∪ [σc, 1 − σc] ∪ [1 − σc, 1], where σc := min{1 4, 2 ε β ln N}. The error bound given above still applies.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Test problem −εuxx + u + ut = 4x(1 − x)et, (x, t) ∈ (0, 1) × (0, 0.5], u(x, 0) = φ(x; ε, d), 0 < x ≤ 1, u(0, t) = 0, u(1, t) = 1, 0 < t ≤ 1,

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.4 0.6 0.8 1 1.2 "Solution for eps=1.d-6"

Figure: Numerical approximation for ε = 10−6 and d = 0.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Parabolic reaction-diffusion problem II

Consider the test problem involving the heat equation ut − εuxx = 0, (x, t) ∈ (0, 1) × (0, 1], and the initial and boundary conditions are taken such that u(x, t) = w0(x, t) = 1 2erf( x 2√εt ), with erf(ζ) = 2 √π ζ exp(−α2) dα is the error function. The initial condition is discontinuous at (0,0), which generates a boundary/corner layer

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Test problem

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5

Figure: Exact solution for ε = 2−10.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Numerical approximation:

We will numerically examine the performance of five approximation methods.

1 Classical scheme on uniform mesh 2 Fitted operator [Hemker and Shishkin, 1994] on uniform mesh 3 Classical operator on fitted mesh applied to regularized

problem.

4 Classical operator on fitted mesh. 5 Fitted operator on fitted mesh.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Classical scheme

On a uniform mesh denoted by QN,M,u we define the classical scheme D−

t UN,M − εδ2 xUN,M = 0,

(x, t) ∈ QN,M,u, UN,M = u, (x, t) ∈ Γ

N,M,u,

Compute the errors and orders of nodal convergence by eN,M

ε

:=

  • UN,M − u
  • ¯

QN,M,u , pN,M ε

:= log2

  • eN,M

ε

/e2N,2M

ε

  • ,

as well as the uniform errors and their orders of convergence eN,M := max

Sε eN,M ε

, pN,M := log2

  • eN,M/e2N,2M

.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Classical scheme

Table: Maximum nodal errors eN,M

ε

and orders of nodal convergence pN,M

ε

using the classical central difference scheme on a uniform mesh.

N=8 N=16 N=32 N=64 N=128 N=256 M=8 M=16 M=32 M=64 M=128 M=256 ε = 20 6.177E-002 6.151E-002 6.179E-002 6.164E-002 6.195E-002 6.199E-002 0.006

  • 0.006

0.003

  • 0.007
  • 0.001

ε = 2−2 2.843E-002 4.877E-002 5.854E-002 5.703E-002 6.111E-002 6.130E-002

  • 0.779
  • 0.263

0.038

  • 0.100
  • 0.005

ε = 2−4 3.131E-002 3.004E-002 2.776E-002 1.647E-002 2.843E-002 4.877E-002 0.060 0.114 0.753

  • 0.788
  • 0.779

ε = 2−6 7.677E-003 2.695E-002 3.263E-002 3.232E-002 3.131E-002 3.004E-002

  • 1.812
  • 0.276

0.014 0.046 0.060 ε = 2−8 4.877E-004 1.945E-003 7.688E-003 2.703E-002 3.288E-002 3.280E-002

  • 1.996
  • 1.983
  • 1.814
  • 0.283

0.004 ε = 2−10 3.052E-005 1.220E-004 4.878E-004 1.945E-003 7.691E-003 2.705E-002

  • 2.000
  • 1.999
  • 1.996
  • 1.983
  • 1.814

ε = 2−12 1.907E-006 7.629E-006 3.052E-005 1.220E-004 4.878E-004 1.945E-003

  • 2.000
  • 2.000
  • 2.000
  • 1.999
  • 1.996

ε = 2−14 1.192E-007 4.768E-007 1.907E-006 7.629E-006 3.052E-005 1.220E-004

  • 2.000
  • 2.000
  • 2.000
  • 2.000
  • 2.000

ε = 2−16 7.451E-009 2.980E-008 1.192E-007 4.768E-007 1.907E-006 7.629E-006

  • 2.000
  • 2.000
  • 2.000
  • 2.000
  • 2.000

ε = 2−18 4.657E-010 1.863E-009 7.451E-009 2.980E-008 1.192E-007 4.768E-007

  • 2.000
  • 2.000
  • 2.000
  • 2.000
  • 2.000

ε = 2−20 2.910E-011 1.164E-010 4.657E-010 1.863E-009 7.451E-009 2.980E-008

  • 2.000
  • 2.000
  • 2.000
  • 2.000
  • 2.000
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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitted operator method [Hemker and Shishkin, 1994]

Define D−

t UN,M − εγ(x, t)δ2 xUN,M = 0,

(x, t) ∈ QN,M,u, UN,M = u, (x, t) ∈ Γ

N,M,u,

where the fitting coefficient γ is chosen such that w0(x, t) = 1 2erf( x 2√εt ) is the exact solution of D−

t w0 − εγ(x, t)δ2 xw0 = 0,

(x, t) ∈ QN,M,u. To reduce rounding effects take γ∗ = D−

t w0+D− t u0

εδ2

xw0+εδ2 xu0

, with u0 = −x3 − 6εxt.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitted operator method

Table: Maximum nodal errors and orders of convergence for the fitted

  • perator method.

N=8 N=16 N=32 N=64 N=128 N=256 M=8 M=16 M=32 M=64 M=128 M=256 ε = 20 2.373E-002 1.042E-002 1.091E-002 7.792E-003 4.772E-003 2.806E-003 1.187

  • 0.066

0.486 0.708 0.766 ε = 2−4 6.076E-003 3.190E-003 1.707E-003 9.029E-004 4.768E-004 2.494E-004 0.930 0.902 0.919 0.921 0.935 ε = 2−8 1.921E-003 7.684E-004 2.831E-004 1.088E-004 4.483E-005 1.992E-005 1.322 1.440 1.380 1.279 1.170 ε = 2−12 1.788E-004 8.376E-005 5.708E-005 2.534E-005 8.668E-006 2.732E-006 1.094 0.553 1.171 1.548 1.666 ε = 2−16 1.118E-005 5.705E-006 2.860E-006 1.326E-006 1.128E-006 6.088E-007 0.971 0.996 1.109 0.234 0.889 ε = 2−20 6.989E-007 3.566E-007 1.787E-007 8.940E-008 4.470E-008 2.910E-008 0.971 0.996 1.000 1.000 0.619 ε = 2−24 4.368E-008 2.229E-008 1.117E-008 5.588E-009 2.794E-009 1.397E-009 0.971 0.996 1.000 1.000 1.000 ε = 2−28 2.730E-009 1.393E-009 6.982E-010 3.492E-010 1.746E-010 8.731E-011 0.971 0.996 1.000 1.000 1.000 ε = 2−30 6.825E-010 3.482E-010 1.746E-010 8.731E-011 4.366E-011 2.183E-011 0.971 0.996 1.000 1.000 1.000 e( ¯ QN,M

u

) 2.373E-002 1.042E-002 1.091E-002 7.792E-003 4.772E-003 2.806E-003 p( ¯ QN,M

u

) 1.187

  • 0.066

0.486 0.708 0.766

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitted operator method

Conclusion: The fitted operator method provides an accurate nodal approximation [Hemker and Shishkin, 1994]. But we cannot conclude that this method provides an accurate global approximation. Consider a very fine mesh ¯ QN,M,u

fine

(with grid points located inside the interior layer) and compute the global errors ¯ UN,M − u ¯

QN,M,u

fine

, where ¯ UN is the linear interpolation of UN defined on the “coarse” uniform mesh ¯ QN,M,u.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Table: The maximum global errors and computed orders of global convergence using the fitted operator method on a uniform mesh.

N=8 N=16 N=32 N=64 N=128 N=256 M=8 M=16 M=32 M=64 M=128 M=256 ε = 20 2.373E-002 1.042E-002 1.106E-002 7.853E-003 4.790E-003 2.813E-003 1.187

  • 0.086

0.494 0.713 0.768 ε = 2−2 1.380E-002 9.135E-003 5.185E-003 2.940E-003 1.627E-003 8.822E-004 0.595 0.817 0.819 0.853 0.883 ε = 2−4 2.636E-002 1.525E-002 7.366E-003 3.750E-003 1.888E-003 9.505E-004 0.790 1.049 0.974 0.990 0.990 ε = 2−6 1.031E-001 4.963E-002 2.396E-002 1.479E-002 7.086E-003 3.707E-003 1.055 1.051 0.696 1.061 0.935 ε = 2−8 2.274E-001 1.726E-001 1.027E-001 4.958E-002 2.390E-002 1.478E-002 0.398 0.748 1.051 1.053 0.693 ε = 2−10 2.500E-001 2.477E-001 2.273E-001 1.725E-001 1.027E-001 4.957E-002 0.014 0.124 0.398 0.748 1.051 ε = 2−12 2.500E-001 2.500E-001 2.500E-001 2.477E-001 2.273E-001 1.725E-001 0.000 0.000 0.013 0.124 0.398 ε = 2−14 2.500E-001 2.500E-001 2.500E-001 2.500E-001 2.500E-001 2.477E-001 0.000 0.000 0.000 0.000 0.013 ε = 2−16 2.500E-001 2.500E-001 2.500E-001 2.500E-001 2.500E-001 2.500E-001 0.000 0.000 0.000 0.000 0.000 ε = 2−18 2.500E-001 2.500E-001 2.500E-001 2.500E-001 2.500E-001 2.500E-001 0.000 0.000 0.000 0.000 0.000

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Regularized problem

An alternative is to use a fitted mesh method with a regularized initial condition: φ∗

ε(x) := 0.5(1 − e−4/√ε)−4(1 − e−4x/√ε)4.

Consider the problem (ureg)t − ε(ureg)xx = 0, (x, t) ∈ (0, 1) × (0, 1], ureg(x, 0) = φ∗

ε(x),

0 < x < 1, ureg(0, t) = 0, ureg(1, t) = u(1, t), t ≥ 0.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Classical operator on Fitted mesh (Regularized problem)

The numerical approximation UN

reg to ureg is generated by using the

classical central difference scheme on a piecewise uniform mesh Q

N,M ∗

which uses N/2 space mesh points in the subintervals [0, σc], [σc, 1], σc := min{1 2, 2 ε β ln N}.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Classical operator on Fitted mesh (Regularized problem)- distribution of error

How are the errors distributed?

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.2 0.4 0.6 0.8 1 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05

Figure: Nodal errors in the region [0, σc/2] with ε = 2−10, N = 64, and M = 64.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Theorem: UN,M

reg

− ureg∞ ≤ CM−1 + C(N−1 ln N)2 + CN−1ε1/2. But, what about ureg − u∞? If σc < 0.5, then can show that u − ¯ UN,M

reg [σc,1]×[0,1] ≤ C(N−1 + M−1).

by using the inequality e−ξ2 ξ +

  • ξ2 + 2

≤ ∞

ξ

e−t2 dt ≤ e−ξ2 ξ +

  • ξ2 + 4/π

.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitted mesh (Regularized problem)- errors outside computational layers

Table: The maximum nodal errors and computed orders of convergence using classical central difference scheme on the piecewise uniform mesh to the regularized problem and computing the errors only on the subregion ¯ QN,M

∗

∩ [σc/2, 1].

N=8 N=16 N=32 N=64 N=128 N=256 M=8 M=16 M=32 M=64 M=128 M=256 ε = 2−2 4.201E-001 4.201E-001 4.201E-001 4.201E-001 4.201E-001 4.201E-001 0.000 0.000 0.000 0.000 0.000 ε = 2−6 5.662E-002 6.923E-002 7.455E-002 7.664E-002 7.767E-002 7.811E-002

  • 0.290
  • 0.107
  • 0.040
  • 0.019
  • 0.008

ε = 2−10 3.204E-002 1.370E-002 6.296E-003 2.835E-003 2.438E-003 2.301E-003 1.225 1.122 1.151 0.218 0.084 ε = 2−14 3.196E-002 1.370E-002 6.296E-003 2.300E-003 5.993E-004 1.221E-004 1.222 1.122 1.453 1.940 2.295 ε = 2−18 3.194E-002 1.370E-002 6.296E-003 2.300E-003 5.993E-004 1.221E-004 1.221 1.122 1.453 1.940 2.295 ε = 2−22 3.194E-002 1.370E-002 6.296E-003 2.300E-003 5.993E-004 1.221E-004 1.221 1.122 1.453 1.940 2.295 ε = 2−26 3.194E-002 1.370E-002 6.296E-003 2.300E-003 5.993E-004 1.221E-004 1.221 1.122 1.453 1.940 2.295 ε = 2−30 3.194E-002 1.370E-002 6.296E-003 2.300E-003 5.993E-004 1.221E-004 1.221 1.122 1.453 1.940 2.295

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Change the test problem ut − εuxx = 0, (x, t) ∈ (0, 1) × (0, 1], where exact solution is u(x, t) = −(x + 0.5)2 − 2εt + 5/2w0(x, t).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0.5 1

Figure: Solution of problem for ε = 2−10, N = 64, and M = 64.

Now for t = 0 we use the values of the exact solution of the parabolic problem. There is no theory (that we know of) currently available.

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Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitted mesh - without regularizing

Table: Maximum two–mesh global differences only on the subregion ¯ QN,M

∗

∩ [√ε ln(1/√ε), 1].

N=8 N=16 N=32 N=64 N=128 N=256 M=8 M=16 M=32 M=64 M=128 M=256 ε = 2−2 5.033E-002 3.002E-002 1.742E-002 1.155E-002 7.255E-003 3.978E-003 0.746 0.785 0.593 0.671 0.867 ε = 2−4 3.076E-002 1.464E-002 8.372E-003 4.362E-003 2.238E-003 1.113E-003 1.071 0.806 0.940 0.963 1.007 ε = 2−6 4.379E-002 1.699E-002 6.682E-003 2.510E-003 1.181E-003 5.375E-004 1.366 1.346 1.413 1.088 1.136 ε = 2−8 3.642E-002 1.499E-002 5.978E-003 2.884E-003 1.020E-003 4.002E-004 1.281 1.326 1.051 1.499 1.350 ε = 2−10 1.758E-002 7.794E-003 4.207E-003 1.985E-003 7.618E-004 3.209E-004 1.174 0.889 1.084 1.382 1.247 ε = 2−12 1.431E-002 4.190E-003 1.569E-003 8.701E-004 3.267E-004 1.635E-004 1.772 1.417 0.850 1.413 0.999 ε = 2−14 1.510E-002 3.575E-003 8.735E-004 3.540E-004 1.310E-004 5.807E-005 2.079 2.033 1.303 1.434 1.174 ε = 2−16 1.540E-002 3.739E-003 9.244E-004 2.285E-004 5.649E-005 1.516E-005 2.042 2.016 2.016 2.016 1.897 ε = 2−18 1.552E-002 3.828E-003 9.503E-004 2.363E-004 5.874E-005 1.460E-005 2.019 2.010 2.008 2.008 2.008 . . . . . . . . . . . . . . . . . . ε = 2−30 1.562E-002 3.905E-003 9.762E-004 2.440E-004 6.100E-005 1.525E-005 2.000 2.000 2.000 2.000 2.000

slide-30
SLIDE 30

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.2 0.4 0.6 0.8 1 −0.04 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04

Figure: Nodal errors in the region [0, σc/2] with ε = 2−10, N = 64, and M = 64 using the fitted mesh method and the discontinuous initial condition.

slide-31
SLIDE 31

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitted operator and fitted mesh

We study a hybrid scheme, which combines the fitted operator method with the piecewise uniform fitted mesh. For t = 0 we use the values of the exact solution of the parabolic problem. There is no theory (that we know of) currently available.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.2 0.4 0.6 0.8 1 −1 1 2 3 4 5 6 7 x 10

−5

Figure: Nodal errors in the region [0, σc/2] with ε = 2−10, N = M = 64.

slide-32
SLIDE 32

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Table: The maximum two–mesh global differences and orders of convergence

N=M=8 N=M=16 N=M=32 N=M=64 N=M=128 N=M=256 ε = 20 2.373E-002 1.042E-002 1.106E-002 7.853E-003 4.790E-003 2.813E-003 1.187

  • 0.086

0.494 0.713 0.768 ε = 2−2 1.380E-002 9.135E-003 5.185E-003 2.940E-003 1.627E-003 8.822E-004 0.595 0.817 0.819 0.853 0.883 ε = 2−4 2.636E-002 1.525E-002 7.366E-003 3.750E-003 1.888E-003 9.505E-004 0.790 1.049 0.974 0.990 0.990 ε = 2−6 1.031E-001 4.963E-002 2.396E-002 1.479E-002 7.086E-003 3.707E-003 1.055 1.051 0.696 1.061 0.935 ε = 2−8 9.645E-002 9.290E-002 7.679E-002 4.958E-002 2.390E-002 1.478E-002 0.054 0.275 0.631 1.053 0.693 ε = 2−10 9.637E-002 9.288E-002 7.570E-002 5.407E-002 3.450E-002 2.003E-002 0.053 0.295 0.486 0.648 0.785 ε = 2−12 9.637E-002 9.288E-002 7.570E-002 5.407E-002 3.450E-002 2.003E-002 0.053 0.295 0.485 0.648 0.785 ε = 2−14 9.636E-002 9.288E-002 7.570E-002 5.407E-002 3.450E-002 2.003E-002 0.053 0.295 0.485 0.648 0.785 ε = 2−16 9.636E-002 9.288E-002 7.570E-002 5.407E-002 3.450E-002 2.003E-002 0.053 0.295 0.485 0.648 0.785 ε = 2−18 9.636E-002 9.288E-002 7.570E-002 5.407E-002 3.450E-002 2.003E-002 0.053 0.295 0.485 0.648 0.785 . . . . . . . . . . . . . . . . . . ε = 2−30 9.636E-002 9.288E-002 7.570E-002 5.407E-002 3.450E-002 2.003E-002 0.053 0.295 0.485 0.648 0.785 ¯ e( ¯ QN,M

∗

) 1.031E-001 9.290E-002 7.679E-002 5.407E-002 3.450E-002 2.003E-002 ¯ p( ¯ QN,M

∗

) 0.151 0.275 0.506 0.648 0.785

slide-33
SLIDE 33

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Fitting factor on Shishkin mesh

Fitting coefficient in the piecewise uniform mesh is defined as: ˆ γ∗(xi, tj) = γ∗(xi, tj), if hi = hi+1, P1[γ∗(xi−1, tj), γ∗(xi+1, tj)],

  • therwise.

0.05 0.1 0.15 0.2 0.25 0.2 0.4 0.6 0.8 1 0.5 1 1.5 1 2 3 4 5 x 10

−3

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Figure: Coefficient ˆ γ∗ in [0, σc] for ε = 2−10 (left) and ε = 2−22 (right).

slide-34
SLIDE 34

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Variable coefficient problem

Consider now the problem 1 + e−2xt 2 ut−εuxx+(1−x)2u = (1−x)(2−x), (x, t) ∈ (0, 1)×(0, 1], where the initial and boundary conditions are u(x, 0) = 1, x ∈ (0, 1), u(0, t) = 0, t ∈ [0, 1], u(1, t) = 1, t ∈ [0, 1]. For its numerical approximation we use the hybrid scheme that uses the finite difference operator 1 + e−2xt 2 D−

t UN,M−εˆ

γ∗δ2

xUN,M+(1−x)2UN,M = (1−x)(2−x), on QN,M ∗

,

  • n the piecewise uniform Shishkin mesh.
slide-35
SLIDE 35

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

0.5 1 0.2 0.4 0.6 0.8 1 −2 −1.5 −1 −0.5 0.5 1 Space variable Time variable 0.5 1 x 10

−3

0.2 0.4 0.6 0.8 1 −1 −0.5 0.5 1 Space variable Time variable

Figure: Numerical solution for ε = 10−8 in the whole domain (left) and a detail of the solution in the layer region (right)

slide-36
SLIDE 36

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Table: Maximum two–mesh global differences and orders of convergence.

N=M=8 N=M=16 N=M=32 N=M=64 N=M=128 N=M=256 ε = 20 5.782E-002 2.995E-002 1.503E-002 7.552E-003 3.765E-003 2.274E-003 0.949 0.995 0.993 1.004 0.728 ε = 2−2 5.206E-002 2.477E-002 1.209E-002 5.972E-003 2.965E-003 1.476E-003 1.071 1.035 1.018 1.010 1.006 ε = 2−4 6.475E-002 2.607E-002 1.138E-002 5.381E-003 2.754E-003 1.405E-003 1.312 1.197 1.080 0.966 0.971 ε = 2−6 2.112E-001 7.310E-002 4.021E-002 2.564E-002 1.271E-002 6.551E-003 1.531 0.862 0.649 1.012 0.957 ε = 2−8 5.959E-001 2.826E-001 1.851E-001 9.266E-002 4.592E-002 2.858E-002 1.076 0.610 0.999 1.013 0.684 ε = 2−10 4.904E-001 3.188E-001 3.432E-001 3.292E-001 2.004E-001 9.753E-002 0.621

  • 0.106

0.060 0.716 1.039 ε = 2−12 4.863E-001 3.176E-001 3.295E-001 3.171E-001 2.654E-001 1.931E-001 0.614

  • 0.053

0.055 0.257 0.459 ε = 2−14 4.847E-001 3.172E-001 3.295E-001 3.171E-001 2.654E-001 1.931E-001 0.612

  • 0.055

0.055 0.257 0.459 ε = 2−16 4.840E-001 3.170E-001 3.295E-001 3.171E-001 2.654E-001 1.931E-001 0.610

  • 0.056

0.055 0.257 0.459 ε = 2−18 4.837E-001 3.170E-001 3.295E-001 3.171E-001 2.654E-001 1.931E-001 0.610

  • 0.056

0.055 0.257 0.459 . . . . . . . . . . . . . . . . . . ε = 2−30 4.835E-001 3.169E-001 3.295E-001 3.171E-001 2.654E-001 1.931E-001 0.609

  • 0.056

0.055 0.257 0.459 d( ¯ QN,M

∗

) 5.959E-001 3.188E-001 3.432E-001 3.292E-001 2.654E-001 1.931E-001 q(QN,M

∗

) 0.903

  • 0.106

0.060 0.311 0.459

slide-37
SLIDE 37

Outline Fitted mesh Fitted operator Regularized Problem Fitted operator and fitted mesh

Conclusions

1 Global accuracy (as opposed to nodal accuracy) is required to

  • btain information within the layers.

2 For singularly perturbed heat equation with a discontinuous

initial condition, fitted operator on uniform mesh generates nodal accuracy but not global accuracy.

3 Computed solution of regularized problem on a fitted mesh

generates global accuracy outside computational layer, for small values of perturbation parameter. Error from regularization persists within the layer.

4 Fitted mesh and classical difference operator outperforms

approximation by regularizing problem. Associated theory to support these observations remains an open question

5 Fitted operator and fitted mesh scheme appears to achieve

global accuracy. Theoretical investigation of the possible convergence of this scheme remains an open question.