Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1s (2025) 

 

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Error Estimates for Three-Dimensional Singularly Perturbed Convection-

Diffusion Problem 

 

Brehmit Kaur1, Vivek Sangwan2 

1Department of Mathematics, Sri Guru Granth Sahib World University, Fatehgarh Sahib 

2School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab, INDIA. 

1Corresponding author: E-mail address: brehmitkaur@yahoo.in, sangwan.vivek@gmail.com 

 

Article History: 

Received: 28-08-2024 

Revised: 09-10-2024 

Accepted: 26-10-2024 

Abstract 

The main aim of the present work is to propose a posteriori error estimates for singularly 

perturbed convection-diffusion problem in three-dimensions. It has been observed that for 

small singular perturbation parameter, the problem under consideration displays nonphysical 

oscillations in the small subregions of the boundary layers. Hughes stabilization strategy 

along with the Streamline upwind/Petrov-Galerkin (SUPG) method has been proposed to 

approximate the solution of the problem. Reliable a posteriori error estimates in energy norm 

on anisotropic meshes have been developed for the proposed method. 

Keywords: 

Convection-diffusion equation, Streamline upwind/Petrov-Galerkin (SUPG) method, Hughes 

stabilization, singularly perturbed problems, a posteriori error estimation, finite element 

method. 

 

1. Introduction 

Many important mathematical models governing various physical phenomena occurring during the 

analysis of biological systems, heat transfer process, mass transfer process, etc., are represented by 

partial differential equations [14,16]. Very few researchers have developed finite element strategies 

for simulating three-dimensional partial differential equations. To name a few, Branco et al. [1] 

proposed three-dimensional finite element technique to analyse the shape evolution of fatigue cracks. 

Numerical tests have been performed and it has been shown that the numerical results agree with the 

experimental findings. Mola et al. [13] developed Streamline upwind Petrov-Galerkin (SUPG) 

technique for approximating unsteady three-dimensional non-linear water waves arising due to ship 

hull advancing in water based on semi-Lagrangian framework. SUPG projection has been considered 

to recover accurate estimates of position vector and potential gradients on free surface. The proposed 

technique results in stabilization of the transport dominated terms and robust adaptation of the spatial 

discretization on unstructured quadrilateral grids. Zhai et al. [17] analysed three-dimensional time 

fractional convection-diffusion equation by proposing an implicit compact finite difference scheme 

which uses fourth-order Pad𝑒 ′ approximation for spatial discretization and central difference scheme 

for time discretization. Mohanty and Setia [11] developed high order compact finite difference 

scheme for approximating three-dimensional quasi-linear elliptic partial differential equation. In the 

present paper, we will focus on proposing the Hughes stabilized SUPG finite element technique for 

solving the singularly perturbed problems.  

mailto:brehmitkaur@yahoo.in


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Consider the three-dimensional SPP given by 

                    −𝛻. 𝜖𝛻𝑢 + 𝒃. 𝛻𝑢 + 𝑐𝑢 = 𝑓        𝑖𝑛   ℾ,                                         (1.1) 

                                                      𝑢 = 0          𝑜𝑛  𝜕ℾ𝐷 ,                                   (1.2) 

 𝜕𝑢

𝜕𝑛
= 𝑔        𝑜𝑛    𝜕ℾ𝑁 ,                                  (1.3) 

where ℾ ⊂ 𝑅3 is a bounded domain with Lipschitz-continuous boundary 𝜕ℾ  and 𝜖 is singular 

perturbation parameter satisfying 0 < 𝜖 ≪ 1. We assume that  𝜕ℾ = 𝜕ℾ𝐷 ∪  𝜕ℾ𝑁  with 𝜕ℾ𝐷  ∩

 𝜕ℾ𝑁  = ∅  , and 𝒃, c and f are analytic.  𝜕ℾ𝐷 and 𝜕ℾ𝑁   represent the Dirichlet and Neumann 

boundaries of the domain respectively. 

The considered problem is comprised of two basic phenomena-convection and diffusion. For the 

case when 𝜖 ≪ ||𝒃||, the above problem becomes convection dominant in nature. This results in 

singularities, such as shocks, interior and boundary layers which deteriorate the accuracy of 

numerical solutions obtained by various numerical schemes. Therefore, it becomes essential to obtain 

some reliable and efficient error estimates for the computed numerical solution to rely on.  

Broadly, error estimates are of two different types, namely, a priori error estimates and a posteriori 

error estimates. It is seen that a priori error estimates provide crude information only about the 

asymptotic behavior of the solution and involves regularity conditions which are very difficult to 

achieve in case of singularities whereas a posteriori error estimates provide quantitative information 

of the computed numerical solution. Therefore, it is more expected to derive some reliable a 

posteriori error estimates based on the data of the problem and the computed numerical solution.   

Stephansen [5] proposed robust a posteriori error estimates for convection-diffusion-reaction 

problems based on weighted interior-penalty discontinuous Galerkin methods. Lazarov [10] derived 

residual based a posteriori estimates for convection-diffusion-reaction equations using finite volume 

element approximations. Carstensen et al. [3] proposed residual-type explicit error estimators and 

averaging techniques for steady convection-diffusion-reaction problems using finite volume method. 

Further, the authors proposed adaptive mesh refining strategy and considered numerical examples to 

test the theoretical findings. 

It has been seen that Streamline upwind/Petrov-Galerkin (SUPG) method provides good approximate 

solution in the region where there is no sharp change in the solution but fails badly in the small 

subregions of sharp boundary layers appearing in the sol. of singularly perturbed problems. It has 

been observed that occurrence of these nonphysical oscillations in the region of sharp boundary 

layers in discrete solution of SUPG method is based on the fact that this scheme is not monotonicity 

preserving. To overcome this hurdle, in the present work, an effort has been made by using Hughes 

stabilization strategy [6] alongwith Streamline upwind/Petrov-Galerkin (SUPG) method. This 

corresponds to addition of one more term in the SUPG discretization of the considered convection-

diffusion problem. A posteriori error estimates have been obtained for the developed technique. 

 

 



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The paper is organised as follows:  

In Section 2, some notations and standard norms have been discussed. Then the variational 

formulation and Hughes stabilized Streamline upwind finite element approximation of the 

continuous problem have been discussed. Section 3 deals with some important tools which are 

essential for deriving reliable error estimates. 

In Section 4, residual based a posteriori error estimates have been derived on anisotropic meshes. In 

the last Section 5, concluding remarks have been made. 

2    Hughes stabilized SUPG technique 

Let 𝑊1,∞(ℾ)and 𝐿∞(ℾ) denote the usual Sobolev space and Lebesgue space respectively. We will use 

the notation (. , .) for inner product (.,.)ℾ. 

We assume that −
1

2
∇. 𝒃 + 𝑐 ≥ 𝑐0  > 0 on ℾ.̅ 

For any open and bounded subset 𝑇 ⊂  ℾ,̅ , let 𝐻1(𝑇) be the standard Sobolev space. Let 

𝑉0 = {𝑤 ∈ 𝐻1(ℾ), 𝑤 = 0  𝑜𝑛 𝜕ℾ𝐷}. 

Since, our objective is to bound the global error 𝑤 − 𝑤ℎ in energy norm, so we define the energy 

norm on any bounded subset 𝑇 ⊂   ℾ̅ as 

|||𝑤|||𝑇 
2 = 𝜖. ||∇𝑤||𝑇

2 + 𝑐0| |𝑤||𝑇
2 .                            (2.1) 

The finite element weak formulation of (1.1) is given as: 

Find 𝑣 ∈ 𝐻1(ℾ ) such that 

𝐵(𝑣, 𝑤) =< 𝐹, 𝑤 > ,                                                (2.2) 

where 

𝐵(𝑣, 𝑤) =  𝜖(∇𝑣, ∇𝑤) + (𝒃. ∇𝑣, 𝑤) + (𝑐𝑣, 𝑤)     (2.3) 

< 𝐹, 𝑤 >= (𝑓, 𝑤) + (𝑔, 𝑤)𝜕𝛺𝑁
, ∀ 𝑤 ∈ 𝑉0. 

The existence and uniqueness of solution of above weak formulation (2.2) can be confirmed using 

the Lax Milgram lemma. Let 𝐹 = {ℾℎ }   denote the family of triangulations of ℾ.  Let ℾℎ  be 

triangulation of domain  ℾ  consisting of tetrahedrons in three-dimensions, assuming only admissible 

and shape-regular triangulation in ℾℎ. For any tetrahedron T with face E, we define 𝑛𝑇,𝐸 = 

(𝑛𝑥 , 𝑛𝑦, 𝑛𝑧) to be the unit outward normal vector for face E of the tetrahedron T. Let 𝑛𝐸  be the 

normal vector for face E obtained from 𝑛𝑇,𝐸  by fixing any of two normal components. 

   2.1 SUPG method 

We define 𝑉ℎ = {𝑤ℎ ∈ 𝐻1 : 𝑤ℎ|𝑇 ∈ 𝑃1(𝑇), ∀ 𝑇 ∈  ℾ ℎ}, where 𝑃1(𝑇) is the space of linear 

polynomials over tetrahedron T and 𝑉0
ℎ = {𝑤ℎ ∈ 𝑉ℎ:  𝑤ℎ|𝜕ℾ𝐷

= 0}. The SUPG method [2] for 

problem (1.1) is given by 

Find 𝑣ℎ ∈ 𝑉ℎ such that 



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𝑩𝜌(𝑣ℎ, 𝑤ℎ) =< 𝐹, 𝑤ℎ >   ∀ 𝑤ℎ ∈ 𝑉0
ℎ,                     (2.1.4) 

where 𝑩𝜌(𝑣ℎ, 𝑤ℎ) = 𝑩(𝑣ℎ, 𝑤ℎ)+ < 𝑅ℎ(𝑣ℎ), 𝜌𝒃. ∇ℎ𝑤ℎ > and < 𝑅ℎ(𝑣ℎ), 𝜌𝒃. ∇ℎ𝑤ℎ >=

∑ 𝜌𝑇𝑇∈ℾ ℎ (−𝜖∆ℎ𝑣ℎ + 𝒃. ∇ℎ𝑣ℎ + 𝑐𝑣ℎ − 𝑓, 𝒃. ∇ℎ𝑤ℎ)𝑇 , 𝜌 is nonnegative stabilization parameter,  

𝑩(𝑣, 𝑤)and <F,w> are defined in (2.3). 

Hughes stabilized SUPG technique 

As is well known that convection dominated problems exhibit nonphysical oscillations at layers as 

𝜖 decreases, the solution of singularly perturbed problem displays boundary layers. Since the 

Streamline upwind/Petrov-Galerkin (SUPG) method results in good approximate solution in the 

region where there is no sharp change in the solution but fails drastically in the subregions of sharp 

boundary layers, in the trial to overcome this hurdle, we propose Hughes stabilization technique [7]  

to SUPG method. It results in an addition of the term <𝑅ℎ(𝑣ℎ), 𝜎𝒃ℎ. ∇ℎ𝑤ℎ > in the SUPG finite 

element discretization of the convection-diffusion equation where 

𝒃ℎ = {

(𝒃. ∇𝒗ℎ)∇𝑣ℎ

|∇𝑣ℎ|2

0,              𝑖𝑓  |∇𝑣ℎ| = 0

, 𝑖𝑓 |∇𝑣ℎ| ≠ 0,    

(2.2.1) 

and 𝜎 is nonnegative stabilization parameter. This additional term increases the robustness of SUPG 

method in the boundary layer region by controlling oscillations. Using Hughes stabilization 

technique to SUPG finite element method, Eq.(1.1) is discretized as follows: 

Find 𝑣ℎ ∈  𝑉ℎ such that  

𝑩𝜌,𝜎(𝑣ℎ, 𝑤ℎ) =< 𝐹, 𝑤ℎ >   ∀ 𝑤ℎ ∈ 𝑉0
ℎ.                                                                (2.2.2) 

Here  𝑩𝜌,𝜎(𝑣ℎ, 𝑤ℎ) = 𝑩(𝑣ℎ, 𝑤ℎ)+< 𝑅ℎ(𝑣ℎ), 𝜌𝒃. ∇ℎ𝑤ℎ > +< 𝑅ℎ(𝑣ℎ), 𝜎𝒃ℎ. ∇ℎ𝑤ℎ > and  <

𝑅ℎ(𝑣ℎ), 𝜎𝒃ℎ. ∇ℎ𝑤ℎ >= ∑ 𝜎𝑇(−𝜖∆ℎ𝑣ℎ𝑇∈ℾℎ + 𝒃. ∇ℎ𝑣ℎ + 𝑐𝑣ℎ − 𝑓, 𝒃ℎ. ∇ℎ𝑤ℎ)𝑇. 

Let 𝜌𝑇  and 𝜎𝑇  be stabilization parameters over each element T. The existence and uniqueness of the 

finite element solution 𝑣ℎ obtained using SUPG finite element discretization has been proved by 

Roos et al. [15]. The stabilization parameter 𝜌𝑇  satisfies 

0 ≤ 𝜌𝑇 ≤
1 

2
min{𝑐0||𝑐||∞,𝑇

−2     ,  (ℎ𝑚𝑖𝑛
𝑇 )−2𝜖−1𝜗−2},                                              (2.2.3) 

where ℎ𝑚𝑖𝑛
𝑇

 is minimal length of element T and the constant 𝜗 satisfies the inequality 

||∇. ∇𝑣ℎ||𝑇 ≤ 𝜗(ℎ𝑚𝑖𝑛
𝑇 )−1||∇𝑣ℎ||𝑇      ∀𝑣ℎ ∈ 𝑉0

ℎ.                                                      (2.2.4) 

It can be observed that  𝜗 = 0 for piecewise linear functions in 𝑉0
ℎ . Therefore, the above bounds 

reduces to 0 ≤ 𝜌𝑇 ≤
𝑐0

2
||𝑐||∞,𝑇

−2 . For easiness, we will use 𝑐 ≾ 𝑑 to denote that there exists a positive 

constant A independent of c,d, ℾ ℎ  and 𝜖 such that 𝑐 ≤ 𝐴𝑑. Further, we assume that 

𝜌𝑇 ≾ ℎ𝑚𝑖𝑛
𝑇 ||𝒃||∞,T

−1  ∀ 𝑇 ∈ ℾ ℎ    .                                                                         (2.2.5).             

                                                          



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3  Some auxiliary tools 

Since the considered singularly perturbed problem displays abrupt changes in the solution when 

Peclet number becomes very large as clear from (3.2), in such situations, elements with large aspect 

ratio i.e. anisotropic meshes are preferred. Therefore, in the present work, anisotropic mesh has been 

considered for domain discretization. In the present Section, some notations on anisotropic meshes 

have been discussed which will be used in later Sections. 

Notations: Consider an arbitrary tetrahedron 𝑇 ∈ ℾ ℎ    with 𝑄0𝑄1as longest edge  (see Fig. 1). 

Represent three orthogonal vectors by 𝑞𝑖,𝑇 with lengths ℎ𝑖,𝑇 = |𝑞𝑖,𝑇|, where 𝑞1,𝑇 is taken along the 

largest edge. From Fig. 1, it can be be confirmed that ℎ1,𝑇 > ℎ2,𝑇 ≥ ℎ3,𝑇. Define ℎ𝑚𝑖𝑛
𝑇 = ℎ3,𝑇. These 

𝑞𝑖,𝑇′𝑠 correspond to three anisotropic directions. Define an orthogonal matrix as 𝐶𝑇 = (𝑞1,𝑇, 𝑞2,𝑇, 𝑞3,𝑇) 

∈ 𝑅3𝑋3. Let 𝛼𝑇   be scaling factor defined as 

𝛼𝑇 = min {𝑐
0

−
1
2, 𝜖−

1
2. ℎ𝑚𝑖𝑛

𝑇 }.                              (3.1) 

We denote tetrahedron by T or T' or 𝑇𝑖, and its faces by E. Denote its height over face E  by 

ℎ𝐸,𝑇 = 3.
|𝑇|

|𝐸|
, 

where |T| represents the volume of the tetrahedron and |E| represents the area of the face E. Let 𝑤𝐸  be 

the bounded domain formed by using two tetrahedrons with common face E and 𝑤𝑇  to be the 

domain consisting of tetrahedron T and its face neighbouring tetrahedra. We denote the local mesh 

Peclet number as 

𝑃𝑒𝑇 =
||𝒃||∞,𝑇 ℎ𝑚𝑖𝑛

𝑇     

2𝜖
,                                (3.2) 

where ℎ𝑚𝑖𝑛
𝑇   is minimal length of element T. Let 

𝑃𝑒𝑤𝑇
= 𝑚𝑎𝑥𝑇′⊂𝑤𝑇

𝑃𝑒𝑇′ 

be mesh Peclet number on the domain 𝑤𝑇. For an interior face E = 𝑇1 ∩ 𝑇2, define face based 

parameters ℎ𝐸 = (ℎ𝐸,𝑇1
+  ℎ𝐸,𝑇2

)/2, ℎ𝑚𝑖𝑛
𝐸

= (ℎ𝑚𝑖𝑛
𝑇1 + ℎ𝑚𝑖𝑛

𝑇2 )/2 + and 𝛼𝐸 = (𝛼𝑇1
+ 𝛼𝑇2

)/2. 

For boundary face E⊂ 𝜕𝑇 ∩ 𝜕ℾ, we define ℎ𝐸 = ℎ𝐸,𝑇, ℎ𝑚𝑖𝑛
𝐸 = ℎ𝑚𝑖𝑛

𝑇
 and 𝛼𝐸 = 𝛼𝑇 .  

We assume 

ℎ𝑖,𝑇~ℎ𝑖,𝑇′    ∀𝑇, 𝑇 ′ with 𝑇 ∩ 𝑇 ′ ≠ ∅, 𝑖 = 1,2,3, 

and number of tetrahedra with node 𝑦𝑗 is bounded uniformly. 

 



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4 Interpolation 

Till now, very few researchers have proposed different a posteriori error estimates on anisotropic 

meshes for three-dimensional singularly perturbed problems. Since the focus is to obtain reliable 

upper error bounds, therefore, a suitable estimate or function called matching function [8] has been 

defined to measure alignment of anisotropic mesh ℾℎ and anisotropic function.  

Matching function: Let  𝑣 ∈ 𝐻1(ℾ) ) and ℾℎ ∈ 𝐹 be triangulation of ℾ . Then 𝑀1: 𝐻
1(ℾ) × 𝐹 → 𝑅 is 

defined as 

𝑀1(𝑣,ℾℎ) ≔ ( ∑ (

𝑇∈ℾℎ

ℎ𝑚𝑖𝑛
𝑇 )−2. |𝐶𝑇

𝑇∇𝑣||𝑇
2 )

1
2/||∇𝑣||,                                    (4.1) 

where 𝐶𝑇 ∈ 𝑅3×3 has been defined earlier.  

In order to derive reliable error estimates in energy norm, we define Cl𝑒 ′ment interpolation operator 

𝐼𝑐 [4] for 𝑣 ∈ 𝐻1(ℾ).  

Lemma 1: Let 𝑣 ∈ 𝐻0
1(ℾ) and 𝛼𝑇 be the scaling factor defined by (3.1). Define Cl𝑒 ′ment 

interpolation operator 𝐼𝑐: 𝐻0
1(ℾ)  → 𝑉0

ℎ as defined in [4, 9]. Then it satisfies 

|||𝐼𝐶𝑣||| ≾ 𝑀1(𝑣,ℾℎ). |||𝑣|||,                                                (4.2) 

∑ 𝛼𝑇
−2

𝑇∈ℾℎ

. ||𝑣 − 𝐼𝑐𝑣||𝑇
2 ≾ 𝑀1(𝑣,ℾℎ)2. |||𝑣|||2,                    (4.3) 

𝜖1/2 ∑ 𝛼𝐸
−1. ||𝑣 − 𝐼𝐶

𝐸⊂ℾ\𝜕ℾ𝐷

𝑣||𝐸
2 ≾ 𝑀1(𝑣,ℾℎ)2. |||𝑣|||2.      (4.4) 

Proof:  The proof of the Lemma has been discussed in [9]. 

4.1 Residual error estimates 

In the present Section, first we will discuss exact and approximate residuals. Then, residual error 

estimator based on residuals has been defined to estimate error in energy norm. Further, reliable error 

bounds for the proposed scheme Hughes stabilized SUPG finite element method has been proposed 

on anisotropic meshes 



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Exact residuals: Let 𝑅𝑇 and 𝑅𝐸 denote the exact element residual and exact face residual over a 

general tetrahedron element T respectively and are defined as 

𝑅𝑇 = 𝑓 − (−𝜖∆𝑢ℎ + 𝒃. ∇𝑢ℎ   + 𝑐𝑢ℎ)         on T, 

 

𝑅𝐸(𝑥) = {

𝑙𝑖𝑚𝑠→0+[𝜕𝑛𝐸
𝑢ℎ(𝑥 + 𝑠𝑛𝐸) − 𝜕𝑛𝐸

𝑢ℎ(𝑥 − 𝑠𝑛𝐸)]       𝑖𝑓 𝐸 ⊂   ℾ\𝜕ℾ,

𝑔 − 𝜕𝑛𝑢ℎ                                                                     𝑖𝑓  𝐸 ⊂ 𝜕ℾ𝑁  ,
0                                                                                     𝑖𝑓  𝐸 ⊂ 𝜕ℾ𝐷 ,

  

where 𝑛𝐸⟂ E is unitary normal vector for face E⊂ ℾ − ℾ𝑁 and n ⟂ E ⊂ 𝜕ℾ𝑁 is outer unitary normal 

vector. 

Approximate residuals: Let Q be approximation operator, used to approximate the element 

residuals and the face residuals i.e. 

                𝑟𝑇 = 𝑄(𝑅𝑇) ∈ 𝑃0(𝑇)        ∀ 𝑇 ∈ ℾℎ, 

𝑟𝐸 = 𝑄(𝑅𝐸) ∈ 𝑃0(𝐸)   ∀ 𝐸. 

where 𝑟𝑇  denotes the approximate element residual and 𝑟𝐸   denotes the approximate face residual. 

Since the numerical solution 𝑢ℎ is linear on E, therefore, 

𝑟𝐸 = 𝑅𝐸    ∀ 𝐸 ⊂ ℾ\𝜕ℾ𝑁. 

Residual error estimator: Residual error estimator 𝑛𝑇  and the approximation term 𝛿𝑇 over any 

tetrahedron T are defined as 

𝑛𝑇
2 = 𝑎𝑇

2 . ||𝑟𝑇||𝑇
2 + 𝜖−

1
2. 𝛼𝑇 . ∑ ||𝑟𝐸

𝐸⊂𝜕𝑇\𝜕ℾ𝐷

||𝐸
2          , 

𝛿𝑇
2 = 𝑎𝑇

2 . ||𝑟𝑇 − 𝑅𝑇||𝑤𝑇
2 + 𝜖−

1
2. 𝛼𝑇 . ∑ ||𝑟𝐸

𝐸⊂𝜕𝑇∩𝜕ℾ𝑁

− 𝑅𝐸||𝐸
2          , 

where 𝛼𝑇 be scaling factor and 𝑎𝑇 = 3𝛼𝑇. Global error estimators are defined as 

Ƞ
2 = ∑ Ƞ

𝑇
2

𝑇∈ℾℎ    and £
2 = ∑ £𝑇

2
𝑇∈ℾℎ    .           (4.1.1) 

Before proposing reliable error upper bounds, we will prove the following two lemmas which will be 

used in the results to follow:  

Lemma 2: Let 𝑣 ∈ 𝐻0
1(ℾ) be exact solution and 𝑣ℎ ∈ 𝑉0

ℎ  be numerical solution obtained by the 

proposed scheme. Then the bilinear form 𝑩𝜌,𝜎(. , . )satisfies the bounds 

𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤 − 𝐼𝑐𝑤) ≾ [( ∑ 𝛼𝑇
2

𝑇∈ℾℎ

||𝑅𝑇||𝑇
2 )

1
2 + ∑ 𝜖−

1
2𝛼𝐸||𝑅𝐸

𝐸⊂ℾ\𝜕ℾ𝐷  

||𝐸
2 )1/2]. 𝑀1(𝑤,ℾℎ). |||𝑤|||. 

Proof: Using Cl𝑒 ′ment interpolation operator 𝐼𝑐, we can write the bilinear form  𝑩𝜌,𝜎(. , . ) as 



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𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤) = 𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤 − 𝐼𝑐𝑤) + 𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝐼𝑐𝑤).         (4.1.2) 

Now, integrating by parts and using exact element residual 𝑅𝑇 and face residual 𝑅𝐸, the bilinear form 

𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤) can be written as 

𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤) = ∑ (𝑅𝑇 , 𝑤)𝑇

𝑇∈ℾℎ

+ ∑ (𝑅𝐸 , 𝑤)𝐸

𝐸⊂ℾ\𝜕ℾ𝐷

 ∀ 𝑤 ∈ 𝐻0
1( ℾ). 

Using the above expression for bilinear form, the middle term of (4.1.2) can be expressed as 

𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤 − 𝐼𝑐𝑤) = ∑ (𝑅𝑇 , 𝑤 − 𝐼𝑐𝑤)𝑇𝑇∈ℾℎ + ∑ (𝑅𝐸 , 𝑤−𝐼𝑐𝑤)𝐸𝐸⊂ℾ\𝜕ℾ𝐷
. 

Using Cauchy Schwarz inequality, we get 

∑ (𝑅𝑇 , 𝑤 − 𝐼𝑐𝑤)𝑇𝑇∈ℾℎ ≤ (∑ 𝛼𝑇
2

𝑇∈ℾℎ ||𝑅𝑇||𝑇
2 )

1

2. (∑ 𝛼𝑇
−2

𝑇∈ℾℎ ||𝑤 − 𝐼𝑐𝑤||𝑇
2 ))

1

2, 

∑ (𝑅𝐸 , 𝑤 − 𝐼𝑐𝑤)𝐸𝐸⊂ℾ\𝜕ℾ𝐷
≤ ((∑ 𝜖−

1

2𝛼𝐸||𝑅𝐸|𝐸⊂ℾ\𝜕ℾ𝐷  |𝐸
2 )

1

2. ( ∑ 𝜖
1

2𝛼𝐸
−1||𝑤 − 𝐼𝑐 𝑤|𝐸⊂ℾ\𝜕ℾ𝐷  |𝐸

2 ))1/2  . 

Further, using Lemma 1, we get 

∑ (𝑅𝑇 , 𝑤 − 𝐼𝑐𝑤)𝑇

𝑇∈ℾℎ

≾ ∑ 𝛼𝑇
2

𝑇∈ℾℎ

||𝑅𝑇||𝑇
2 )

1
2. 𝑀1(𝑤,ℾℎ). |||𝑤|||, 

∑ (𝑅𝐸 , 𝑤 − 𝐼𝑐𝑤)𝐸𝐸⊂ℾ\𝜕ℾ𝐷
≾ ( ∑ 𝜖−

1

2𝛼𝐸||𝑅𝐸𝐸⊂ℾ\𝜕ℾ𝐷  ||𝐸
2 )1/2.𝑀1(𝑤, ℾℎ)). |||𝑤|||. 

Therefore, the term 𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤 − 𝐼𝑐𝑤) is bounded above by 

𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝑤 − 𝐼𝑐𝑤) ≾ [ ∑ 𝛼𝑇
2

𝑇∈ℾℎ

||𝑅𝑇||𝑇
2 )

1
2 + ∑ 𝜖−

1
2𝛼𝐸||𝑅𝐸|

𝐸⊂ℾ\𝜕ℾ𝐷  

|𝐸
2 )

1
2]. 𝑀1(𝑤,ℾℎ)). |||𝑤|||. 

Lemma 3:   Let 𝑣 ∈ 𝐻0
1(ℾ)  be exact  solution  and 𝑣ℎ ∈ 𝑉0

ℎ be the approximate solution. Then 

Cl𝑒 ′ment interpolation operator 𝐼𝐶: 𝐻0
1(ℾ)  → 𝑉0

ℎ  satisfies the inequality 

𝑩𝜌,𝜎(𝑣 − 𝑣ℎ, 𝐼𝑐𝑤) ≾ 2( ∑ 𝛼𝑇
2

𝑇∈ℾℎ

||𝑅𝑇||𝑇
2 )

1
2. 𝑀1(𝑤,ℾℎ). |||𝑤|||. 

Proof: For any mesh function 𝑣ℎ ∈ 𝑉0
ℎ, using (2.2.4) and scaling arguments, we can get 

||∇𝑣ℎ||𝑇 ≤ (ℎ𝑚𝑖𝑛
𝑇 )−1||𝑣ℎ||𝑇. 

From energy norm def.(2.1), we have 

||𝑣ℎ||𝑇 ≤ 𝑐
0

−
1

2|||𝑣ℎ|||𝑇. 

→ ||∇𝑣ℎ||𝑇 ≾ (ℎ𝑚𝑖𝑛
𝑇 )−1𝑐

0

−
1

2|||𝑣ℎ|||𝑇.                        (4.1.3) 

Again, from energy norm, we get 

||∇𝑣ℎ||𝑇 ≤ 𝜖−1/2|||𝑣ℎ|||𝑇 .                                         (4.1.4) 

 



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→ ||∇𝑣ℎ||𝑇 ≾ min {(ℎ𝑚𝑖𝑛
𝑇 )−1𝑐

0

−
1

2, 𝜖−
1

2}|||𝑣ℎ|||𝑇.     (4.1.5) 

On simplification, inequality (4.1.5) reduces to 

||∇𝑣ℎ||𝑇 ≾ min {(ℎ𝑚𝑖𝑛
𝑇 )−1𝑐

0

−
1

2, 𝜖−
1

2}|||𝑣ℎ|||𝑇  =  (ℎ𝑚𝑖𝑛
𝑇 )−1𝛼𝑇||| 𝑣ℎ|||𝑇 𝑓𝑜𝑟  𝑣ℎ ∈ 𝑉0

ℎ.   

                         (Using Eq.(3.1)) 

𝐵𝜌,𝜎(𝑣 − 𝑣ℎ, 𝐼𝑐𝑤) =< 𝜖∇𝑣. ∇𝐼𝑐𝑤 >  + < 𝒃. ∇𝑣, 𝐼𝑐𝑤 > + < 𝑐𝑣, 𝐼𝑐𝑤 > -{< 𝜖∇𝑣ℎ, ∇𝐼𝑐𝑤> + 

<𝒃. ∇𝑣ℎ, 𝐼𝑐𝑤 >  +< 𝑐𝑣ℎ, 𝐼𝑐𝑤 >  + ∑ 𝜌𝑇𝑇 (𝑅𝑇 , 𝒃. ∇𝐼𝑐𝑤) + ∑ 𝜎𝑇𝑇 (𝑅𝑇 , 𝒃𝒉. ∇𝐼𝑐𝑤)}. 

 

Now based on the Galerkin orthogonal property and standard scaling results, the above equation can 

be rewritten as  

𝐵𝜌,𝜎(𝑣 − 𝑣ℎ, 𝐼𝑐𝑤) = − ∑ 𝜌𝑇

𝑇

(𝑅𝑇 , 𝒃. ∇𝐼𝑐𝑤) − ∑ 𝜎𝑇

𝑇

(𝑅𝑇 , 𝒃𝒉. ∇𝐼𝑐𝑤) 

≤  ∑ 𝜌𝑇||𝑅𝑇

𝑇∈ℾℎ

||𝑇||𝒃||∞,𝑇||∇𝐼𝑐𝑤||𝑇    +  ∑ 𝜎𝑇||𝑅𝑇

𝑇∈ℾℎ

||𝑇||𝒃𝒉||∞,𝑇||∇𝐼𝑐𝑤||𝑇     

                              

≤  ∑ 𝜌𝑇||𝑅𝑇

𝑇∈ℾℎ

||𝑇||𝒃||∞,𝑇(ℎ𝑚𝑖𝑛
𝑇 )−1𝛼𝑇|||𝐼𝑐𝑤|||𝑇

+  ∑ 𝜎𝑇||𝑅𝑇

𝑇∈ℾℎ

||𝑇||𝒃𝒉||∞,𝑇 (ℎ𝑚𝑖𝑛
𝑇 )−1𝛼𝑇|||𝐼𝑐𝑤|||𝑇 . 

Using Lemma 1, we get 

|||𝐼𝑐𝑤||| ≤ 𝑀1(𝑤,ℾℎ). |||𝑤||| ∀  𝑤 ∈ 𝐻0
1(ℾ). 

Thus, we have 

𝐵𝜌,𝜎(𝑣 − 𝑣ℎ, 𝐼𝑐𝑤) ≤ ( ∑ 𝜌𝑇
2

𝑇∈ℾℎ

||𝑅𝑇||2||𝒃||∞,𝑇
2 (ℎ𝑚𝑖𝑛

𝑇 )−2𝛼𝑇
2 )

1
2. 𝑀1(𝑤,ℾℎ). |||𝑤||| 

                                              +(∑ 𝜎𝑇
2

𝑇∈ℾℎ ||𝑅𝑇||2||𝒃𝒉||∞,𝑇
2 (ℎ𝑚𝑖𝑛

𝑇 )−2𝛼𝑇
2 )

1

2. 𝑀1(𝑤,ℾℎ). |||𝑤|||. 

It may be noted that the effect of nonlinear term 𝒃ℎ in the 𝐿∞ norm will be bounded by that of the 

term ||𝒃||∞ as shown below i.e. 

𝒃𝒉 =
(𝒃. ∇𝑣ℎ)∇𝑣ℎ

|∇𝑣ℎ|𝟐
,   |∇𝑣ℎ| ≠ 0 

𝒃𝒉 ≤
||𝒃||||∇𝑣ℎ||∇𝑣ℎ

|∇𝑣ℎ|𝟐
   {𝑈𝑠𝑖𝑛𝑔 𝐶𝑎𝑢𝑐ℎ𝑦 − 𝑆𝑐ℎ𝑤𝑎𝑟𝑧 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦} 



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||𝒃𝒉|| ≤
||𝒃||||𝛁𝒗𝒉||||𝛁𝒗𝒉||

|∇𝑣ℎ|𝟐
 

≾ ||𝒃|| 

From relation (2.2.5), we get 

𝜌𝑇 ≾ ℎ𝑚𝑖𝑛
𝑇 /||𝒃||∞,𝑇       ∀ 𝑇 ∈ ℾℎ, 

𝜎𝑇 ≾ ℎ𝑚𝑖𝑛
𝑇 /||𝒃𝒉||∞,𝑇       ∀ 𝑇 ∈ ℾℎ. 

Therefore, 

𝐵𝜌,𝜎(𝑣 − 𝑣ℎ, 𝐼𝑐𝑤) ≾ [( ∑ 𝛼𝑇
2 |

𝑇∈ℾℎ

|𝑅𝑇||𝑇
2 )

1
2   +  ( ∑ 𝛼𝑇

2 |

𝑇∈ℾℎ

|𝑅𝑇||𝑇
2 )

1
2]. 𝑀1(𝑤,ℾℎ). |||𝑤||| 

                                               ≾ 2(∑ 𝛼𝑇
2 |𝑇∈ℾℎ

|𝑅𝑇||𝑇
2 )

1

2. 𝑀1(𝑤, ℾℎ). |||𝑤|||. 

Theorem: (Residual error estimation) Let 𝑣 ∈ 𝐻0
1(ℾ) be the exact solution and 𝑣ℎ ∈ 𝑉0

ℎ be the 

Hughes stabilized SUPG finite element solution of (1.1)-(1.3). Then the error in energy norm is 

bounded above globally by 

|||𝑣 − 𝑣ℎ||| ≤ 𝑀1(𝑣 − 𝑣ℎ, ℾℎ). [Ƞ + £]. 

Proof: We know that 𝑩𝜌,𝜎(𝑣, 𝑣) ≥ |||𝑣|||2    ∀𝑣 ∈ 𝐻0
1(ℾ). Using this result, we get 

|||𝑣 − 𝑣ℎ||| ≤
𝐵𝜌,𝜎( 𝑣−𝑣ℎ       ) 

|||𝑣−𝑣ℎ|||
,                                                                     (4.1.6) 

where 𝑤 = 𝑣 − 𝑣ℎ. Putting 𝑎𝑇 = 3𝛼𝑇 and using Lemma 2 and Lemma 3 in Eq.(4.1.2), 

the above equation reduces to  

|||𝑣 − 𝑣ℎ||| ≤ [(∑ 𝛼𝑇
2 ||𝑅𝑇𝑇∈ℾℎ ||𝑇

2 )
1

2 + (∑ 𝜖−
1

2𝛼𝐸||𝑅𝐸𝐸⊂ℾ\𝜕ℾ𝐷  ||𝐸
2 )1/2]. 𝑀1(𝑤,ℾℎ).       

(4.1.7) 

Using triangle inequalities  

||𝑅𝑇||𝑇
2  ≤ ||𝑟𝑇 − 𝑅𝑇||𝑇

2 + ||𝑟𝑇||𝑇
2  , 

||𝑅𝐸||𝑇
2  ≤ ||𝑟𝐸 − 𝑅𝐸||𝐸

2 + ||𝑟𝐸||𝐸 
2 , 

Substituting these inequalities in Eq.(4.1.7), we get 

|||𝑣 − 𝑣ℎ||| ≤ 𝑀1(𝑣 − 𝑣ℎ, ℾℎ). [Ƞ + £], 

where Ƞ
2
 and £

2
 are defined in Eq.(4.1.1). 

5   Conclusion 

In the presented work, reliable a posteriori error estimates have been proposed for singularly 

perturbed convection-diffusion problem in three-dimensions. Hughes stabilization technique under 



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SUPG finite element framework has been proposed to approximate the solution of singularly 

perturbed problems. Anisotropic meshes have been considered for the domain discretization. At the 

end, residual based error estimates in energy norm on anisotropic meshes have been developed. 

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