Functional A Posteriori Error Control for Conforming Mixed Approximations
of Coercive Problems with Lower Order Terms

Immanuel Anjam; Dirk Pauly

doi:10.1515/cmam-2016-0016

Article

Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

Immanuel Anjam and Dirk Pauly

Published/Copyright: May 9, 2016

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From the journal Computational Methods in Applied Mathematics Volume 16 Issue 4

Abstract

The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class A*⁢A⁢x+x=f or in mixed formulation A*⁢y+x=f, A⁢x=y, where the exact solution (x,y) is in D⁢(A)×D⁢(A*). Here A is a linear, densely defined and closed (usually a differential) operator and A* its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation (x~,y~) belongs to D⁢(A)×D⁢(A*). In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality

|x-x~|2+|A⁢(x-x~)|2+|y-y~|2+|A*⁢(y-y~)|2=ℳ⁢(x~,y~),

where ℳ⁢(x~,y~):=|f-x~-A*⁢y~|2+|y~-A⁢x~|2 contains only known data. Our second main result is an error estimate for all equations of the class A*⁢A⁢x+i⁢x=f or in mixed formulation A*⁢y+i⁢x=f, A⁢x=y, where i is the imaginary unit. For this problem we have the two-sided estimate

22+1⁢ℳi⁢(x~,y~)≤|x-x~|2+|A⁢(x-x~)|2+|y-y~|2+|A*⁢(y-y~)|2≤22-1⁢ℳi⁢(x~,y~),

where ℳi⁢(x~,y~):=|f-i⁢x~-A*⁢y~|2+|y~-A⁢x~|2 contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

Keywords: Functional A Posteriori Error Estimates; Error Equalities; Mixed Formulations; Combined Norm

MSC 2010: 65N15

Dedicated to Sergey Igorevich Repin

Funding statement: The first author thanks the Emil Aaltonen Foundation for support.

A Inhomogeneous and More Boundary Conditions

We will demonstrate that our results also hold for Robin type boundary conditions, which means that our results are true for many commonly used boundary conditions. Moreover, we emphasize that we can also handle inhomogeneous boundary conditions. Since it is clear that this method works in the general setting for both Cases I and II, we will demonstrate it here just for a simple reaction-diffusion type model problem belonging to the class of Case I. Let Ω be as in the latter section and now the boundary Γ be decomposed into three disjoint parts Γ𝙳, Γ𝙽 and Γ𝚁.

The model problem is: Find the scalar potential u∈𝖧1 such that

{-div⁡∇⁡u+u=fin ⁢Ω,u=g1on ⁢Γ𝙳,n⋅∇⁡u=g2on ⁢Γ𝙽,n⋅∇⁡u+γ⁢u=g3on ⁢Γ𝚁

hold. Hence, on Γ𝙳,Γ𝙽 and Γ𝚁 we impose Dirichlet, Neumann and Robin type boundary conditions, respectively. In the Robin boundary condition, we assume that the coefficient γ≥γ0>0 belongs to 𝖫∞. The dual variable for this problem is the flux p:=∇⁡u∈𝖣. Furthermore, as long as Γ𝚁≠∅ and to avoid tricky discussions about traces and the corresponding 𝖧-1/2-spaces of Γ, Γ𝙳,Γ𝙽, and Γ𝚁, which can be quite complicated, we assume for simplicity that u∈𝖧2. Then, p∈𝖧1 and all gi belong to 𝖫2 even to 𝖧1/2 of Γ. For the norms we simply have

∥(u,p)∥2=|u|𝖧12+|p|𝖣2.

Theorem A.1

For any approximation pair (u~,p~)∈H2×H1 with u-u~∈HΓD1 and p-p~∈DΓN as well as n⋅(p-p~)+γ⁢(u-u~)=0 on ΓR it holds

∥(u,p)-(u~,p~)∥2+2⁢|u-u~|𝖫2⁢(Γ𝚁),γ2=ℳ⁢(u~,p~)

with M⁢(u~,p~):=|f-u~+div⁡p~|L22+|p~-∇⁡u~|L22. Moreover, |u-u~|L2⁢(ΓR),γ=|n⋅(p-p~)|L2⁢(ΓR),γ-1.

Proof.

Following the proof of Theorem 2.5, we have

ℳ(u~,p~)=|u-u~|𝖧12+|p-p~|𝖣2⏟=∥(u,p)-(u~,p~)∥2+2ℜ〈u-u~,div(p~-p)〉𝖫2+2ℜ〈∇(u-u~),p~-p〉𝖫2.

Moreover, since n⋅(p~-p) and u-u~ belong to 𝖫2⁢(Γ), we have

〈∇⁡(u-u~),p~-p〉𝖫2+〈u-u~,div⁡(p~-p)〉𝖫2=〈n⋅(p~-p),u-u~〉𝖫2⁢(Γ)

=〈n⋅(p~-p),u-u~〉𝖫2⁢(Γ𝚁)

=〈γ⁢(u-u~),u-u~〉𝖫2⁢(Γ𝚁).

As 〈γ⁢(u-u~),u-u~〉𝖫2⁢(Γ𝚁)=〈γ-1⁢n⋅(p-p~),n⋅(p-p~)〉𝖫2⁢(Γ𝚁), we get the assertion. ∎

Remark A.2

If all gi=0, we can set (u~,p~)=(0,0) and get

∥(u,p)∥2+2⁢|u|𝖫2⁢(Γ𝚁),γ2=|f|𝖫22,

which follows also by

|f|𝖫22=|divp|𝖫22+|u|𝖫22-2ℜ〈div∇u,u〉𝖫2

=|divp|𝖫22+|u|𝖫22+2|∇u|𝖫2-2ℜ〈n⋅∇u,u〉𝖫2⁢(Γ)

=|div⁡p|𝖫22+|u|𝖫22+2⁢|∇⁡u|𝖫2-2⁢ℜ⁡〈n⋅∇⁡u,u〉𝖫2⁢(Γ𝚁)⏟=-|u|𝖫2⁢(Γ𝚁),γ2.

Thus, in this case the assertion of Theorem A.1 has a normalized counterpart as well.

If Γ𝚁=∅, we have a pure mixed Dirichlet and Neumann boundary.

Theorem A.3

Let ΓR=∅. For any approximation (u~,p~)∈H1×D with u-u~∈HΓD1 and p-p~∈DΓN we have

∥(u,p)-(u~,p~)∥2=ℳ⁢(u~,p~).

Corollary A.4

Let ΓR=∅. Theorem A.3 provides the well-known a posteriori error estimates for the primal and dual problems.

For any u~∈𝖧1 with u-u~∈𝖧Γ𝙳1 it holds
|u-u~|𝖧12=minψ∈𝖣p-ψ∈𝖣Γ𝙽⁡ℳ⁢(u~,ψ)=ℳ⁢(u~,p).
For any p~∈𝖣 with p-p~∈𝖣Γ𝙽 it holds
|p-p~|𝖣2=minφ∈𝖧1u-φ∈𝖧Γ𝙳1⁡ℳ⁢(φ,p~)=ℳ⁢(u,p~).

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Received: 2015-12-29

Revised: 2016-3-16

Accepted: 2016-3-18

Published Online: 2016-5-9

Published in Print: 2016-10-1

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Articles in the same Issue

https://doi.org/10.1515/cmam-2016-0016

Keywords for this article

Functional A Posteriori Error Estimates; Error Equalities; Mixed Formulations; Combined Norm