In this study, a wavelet multi-resolution interpolation Galerkin method (WMIGM) is proposed to solve linear singularly perturbed boundary value problems. Unlike conventional wavelet schemes, the proposed algorithm can be readily extended to special node generation techniques, such as the Shishkin node. Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients. All the shape functions possess the Kronecker delta property, making the imposition of boundary conditions as easy as that in the finite element method. Four numerical examples are studied to demonstrate the validity and accuracy of the proposed wavelet method. The results show that the use of modified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer. Compared with many other methods, the proposed method possesses satisfactory accuracy and efficiency. The theoretical and numerical results demonstrate that the order of the

The singularly perturbed boundary value problem originates from fluid mechanics and arises in the mathematical modeling of physical engineering problems. In this study, we consider the following second-order singularly perturbed two-point boundary value problem:

where

The presence of boundary layers makes it challenging to solve singularly perturbed problems using classical numerical methods. Consequently, numerous special schemes have been proposed, including transforming the boundary value problem into an initial value problem [

Over recent decades, meshfree methods that rely only on nodes to make approximations have attracted considerable research interest [

Wavelet-based methods have gained increasing attention as they are widely and successfully used in practical applications [

Recently, we proposed a truly meshfree method based on wavelet multiresolution analysis, known as the wavelet multiresolution interpolation Galerkin method (WMIGM) [

The proposed wavelet multiresolution interpolation formula possesses the Kronecker delta function property;

Polynomial reproduction can be achieved up to

It does not require matrix inversions or ad-hoc parameters;

The stiffness matrix can be efficiently obtained with an analytical integration method.

A key challenge to solving singularly perturbed boundary value problems is dealing with their singularity, which can lead to numerical instability, oscillations, and spurious solutions [

This paper is organized as follows. In

This section reviews the construction of the wavelet multiresolution interpolation approximation and explains several key properties of the wavelet interpolating shape function that was proposed in our previous works [

We begin with the interpolating wavelet transform [

where

is yielded, where

Next, we focus on the interpolating wavelet transform construction for interval

where

By applying

Based on the compact support of the scaling function

It can be seen from

in which

Substituting

where the modified wavelet scaling function

Compact support:

Interpolation:

Polynomial reproduction:

Notice that the grid points in

where

with the following initial condition:

In

Interpolation:

Polynomial reproduction:

where

In this section, we formulate the WMIGM for singularly perturbed boundary value problems on Shishkin nodes [

We first divide the computational domain

in which

In the subsequent calculations, we set

We then evenly arrange the nodes into two subintervals:

for left-side boundary layer problems, and

for right-side boundary layer problems.

After the nodes are generated, we use the iterative formula in

The variational form of

Then, we replace the infinite dimensional space,

where

Using

Substituting

in which the matrices are

Using this process, we can obtain the WMIGM solutions by solving

In this section, we provide the WMIGM’s error estimate, which relies on the modified Shishkin nodes. In the following analysis, we assume that

Based on the properties of the wavelet multiresolution interpolating shape functions introduced in the previous

where

where

Then, from

Obviously,

Thus, it remains to prove the estimate

The estimation of

in which

Owing to the Hölder inequalities and

With the aid of integrating by parts and the Cauchy-Schwarz inequality, we observe that

As a result,

Hence,

which is the required result.

In this section, we applied the WMIGM to four examples to evaluate its numerical accuracy. We considered two grid points: uniform node points using the wavelet interpolation Galerkin method (WIGM) and refined local grid points using the WMIGM, as specified in

To estimate the accuracy of the solutions, the maximum absolute and

and the numerical rates of convergence are

All numerical results were conducted on an AMD Ryzen 7 3700X CPU @ 3.20 GHz with 64 GB RAM in MATLAB.

We begin our numerical cases with the following left-side boundary layer problem [

The boundary conditions are extracted from the exact solution as

The absolute errors obtained by the proposed WIGM with _{max} is approximately

Method | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{max} |
_{max} |
_{max} |
_{max} |
_{max} |
|||||||

16 | WIGM | 3.53E-05 | 1.06E-03 | 1.53E-02 | 8.65E-02 | ||||||

WMIGM | 3.53E-05 | 1.68E-04 | 4.47E-03 | 1.56E-02 | |||||||

32 | WIGM | 9.04E-07 | 5.29 | 4.12E-05 | 4.69 | 1.14E-03 | 3.75 | 1.58E-02 | 2.46 | 8.75E-02 | |

WMIGM | 9.04E-07 | 5.29 | 5.01E-06 | 5.06 | 4.09E-05 | 6.77 | 5.11E-03 | 1.61 | 1.60E-02 | ||

64 | WIGM | 1.82E-08 | 5.64 | 1.05E-06 | 5.29 | 4.41E-05 | 4.69 | 1.17E-03 | 3.75 | 1.60E-02 | 2.45 |

WMIGM | 1.82E-08 | 5.64 | 1.09E-07 | 5.52 | 6.79E-07 | 5.91 | 3.96E-05 | 7.01 | 5.48E-03 | 1.55 | |

NOBS [ |
7.43E-10 | 5.21E-08 | 2.91E-06 | 1.15E-04 | 1.55E-03 | ||||||

QBSM [ |
9.27E-08 | 1.71E-06 | 2.75E-05 | 3.78E-04 | 5.38E-03 | ||||||

128 | WIGM | 3.22E-10 | 5.82 | 2.12E-08 | 5.64 | 1.13E-06 | 5.29 | 4.56E-05 | 4.69 | 1.19E-03 | 3.75 |

WMIGM | 3.22E-10 | 5.82 | 2.02E-09 | 5.76 | 1.25E-08 | 5.77 | 6.31E-07 | 5.97 | 3.79E-05 | 7.18 | |

NOBS [ |
1.18E-11 | 5.97 | 8.71E-10 | 5.90 | 5.58E-08 | 5.70 | 3.00E-06 | 5.26 | 1.17E-04 | 3.73 | |

QBSM [ |
5.81E-09 | 3.99 | 1.09E-07 | 3.98 | 1.83E-06 | 3.91 | 2.85E-05 | 3.73 | 3.84E-04 | 3.81 | |

256 | WIGM | 5.46E-12 | 5.88 | 3.76E-10 | 5.82 | 2.27E-08 | 5.64 | 1.17E-06 | 5.29 | 4.63E-05 | 4.69 |

WMIGM | 5.46E-12 | 5.88 | 3.45E-11 | 5.87 | 2.21E-10 | 5.82 | 1.16E-08 | 5.77 | 5.87E-07 | 6.01 | |

NOBS [ |
1.86E-13 | 5.99 | 1.39E-11 | 5.97 | 9.33E-10 | 5.90 | 5.77E-08 | 5.7 | 3.05E-06 | 5.26 | |

QBSM [ |
3.64E-10 | 4.00 | 6.81E-09 | 3.99 | 1.16E-07 | 3.98 | 1.89E-06 | 3.91 | 2.89E-05 | 3.73 | |

512 | WIGM | - | - | 1.11E-11 | 5.08 | 4.04E-10 | 5.81 | 2.35E-08 | 5.64 | 1.19E-06 | 5.29 |

WMIGM | - | - | 3.68E-12 | 5.91 | 2.07E-10 | 5.81 | 1.08E-08 | 5.77 | |||

NOBS [ |
2.91E-15 | 6.00 | 2.18E-13 | 5.99 | 1.45E-11 | 5.97 | 9.64E-10 | 5.90 | 5.86E-08 | 5.70 | |

QBSM [ |
2.27E-11 | 4.00 | 4.26E-10 | 4.00 | 7.30E-09 | 3.99 | 1.20E-07 | 3.98 | 1.92E-06 | 3.91 | |

1024 | WIGM | - | - | - | - | - | - | 4.18E-10 | 5.81 | 2.38E-08 | 5.64 |

WMIGM | - | - | - | - | - | - | - | - | 1.91E-10 | 5.81 | |

NOBS [ |
4.54E-17 | 6.00 | 3.41E-15 | 6.00 | 2.33E-13 | 5.99 | 1.55E-11 | 5.96 | 9.80E-10 | 5.90 | |

QBSM [ |
1.35E-12 | 4.07 | 2.65E-11 | 4.01 | 4.57E-10 | 4.00 | 7.54E-09 | 3.99 | 1.22E-07 | 3.98 |

We next consider a source-free singularly perturbed problem as described in [

whose analytical solution is given by

in which

Method | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{max} |
R | _{max} |
R | _{max} |
R | _{max} |
R | _{max} |
R | ||

16 | WIGM | 4.71E-05 | 9.30E-04 | 1.09E-02 | 5.64E-02 | ||||||

WMIGM | 4.71E-05 | 1.53E-04 | 2.98E-03 | 1.02E-02 | |||||||

32 | WIGM | 1.28E-06 | 5.21 | 3.82E-05 | 4.61 | 8.48E-04 | 3.68 | 1.06E-02 | 2.41 | 5.63E-02 | |

WMIGM | 1.28E-06 | 5.21 | 4.74E-06 | 5.01 | 2.65E-05 | 6.81 | 3.32E-03 | 1.62 | 1.04E-02 | ||

64 | WIGM | 2.64E-08 | 5.59 | 1.01E-06 | 5.25 | 3.39E-05 | 4.65 | 8.07E-04 | 3.71 | 1.04E-02 | 2.43 |

WMIGM | 2.64E-08 | 5.59 | 1.06E-07 | 5.49 | 4.38E-07 | 5.92 | 2.51E-05 | 7.04 | 3.51E-03 | 1.57 | |

NOBS [ |
9.76E-10 | 4.73E-08 | 2.16E-06 | 5.10E-05 | 1.03E-03 | ||||||

QBSM [ |
1.02E-07 | 1.42E-06 | 1.99E-05 | 2.55E-04 | 3.55E-03 | ||||||

128 | WIGM | 4.75E-10 | 5.80 | 2.05E-08 | 5.61 | 8.81E-07 | 5.27 | 3.18E-05 | 4.67 | 7.87E-04 | 3.73 |

WMIGM | 4.75E-10 | 5.80 | 1.97E-09 | 5.74 | 8.08E-09 | 5.76 | 3.98E-07 | 5.98 | 2.39E-05 | 7.20 | |

NOBS [ |
1.57E-11 | 5.96 | 7.97E-10 | 5.89 | 4.24E-08 | 5.90 | 2.06E-06 | 4.63 | 7.66E-05 | 3.74 | |

QBSM [ |
6.41E-09 | 3.99 | 9.01E-08 | 3.98 | 1.33E-06 | 3.91 | 1.92E-05 | 3.73 | 2.50E-04 | 3.82 | |

256 | WIGM | 1.19E-11 | 5.32 | 3.67E-10 | 5.81 | 1.78E-08 | 5.63 | 8.20E-07 | 5.28 | 3.08E-05 | 4.68 |

WMIGM | 1.19E-11 | 5.32 | 3.35E-11 | 5.88 | 1.45E-10 | 5.80 | 7.33E-09 | 5.76 | 3.70E-07 | 6.02 | |

NOBS [ |
2.47E-13 | 5.99 | 1.27E-11 | 5.97 | 7.11E-10 | 5.97 | 4.00E-08 | 5.69 | 2.01E-06 | 5.25 | |

QBSM [ |
4.01E-10 | 4.00 | 5.67E-09 | 3.99 | 8.43E-08 | 3.98 | 1.28E-06 | 3.91 | 1.89E-05 | 3.73 | |

512 | WIGM | - | - | - | - | 3.16E-10 | 5.82 | 1.66E-08 | 5.63 | 7.91E-07 | 5.28 |

WMIGM | - | - | - | - | 8.36E-12 | 4.12 | 1.33E-10 | 5.78 | 6.78E-09 | 5.77 | |

NOBS [ |
3.86E-15 | 6.00 | 2.00E-13 | 5.99 | 1.13E-11 | 5.97 | 6.70E-10 | 5.90 | 3.88E-08 | 5.70 | |

QBSM [ |
2.51E-11 | 4.00 | 3.55E-10 | 4.00 | 5.29E-09 | 3.99 | 8.14E-08 | 3.98 | 1.26E-06 | 3.91 | |

1024 | WIGM | - | - | - | - | - | - | 2.91E-10 | 5.83 | 1.59E-08 | 5.63 |

WMIGM | - | - | - | - | - | - | - | - | 1.26E-10 | 5.75 | |

NOBS [ |
6.03E-17 | 6.00 | 3.13E-15 | 6.00 | 1.78E-13 | 5.99 | 1.07E-11 | 5.97 | 6.49E-10 | 5.90 | |

QBSM [ |
1.74E-12 | 3.85 | 2.22E-11 | 4.00 | 3.31E-10 | 4.00 | 5.11E-09 | 3.99 | 7.99E-08 | 3.98 |

We next consider the right-side boundary layer problem [

with boundary conditions of

Method | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{max} |
R | _{max} |
R | _{max} |
R | _{max} |
R | _{max} |
R | ||

16 | WIGM | 4.71E-05 | 1.21E-03 | 1.63E-02 | 8.92E-02 | ||||||

WMIGM | 4.71E-05 | 1.92E-04 | 4.70E-03 | 1.57E-02 | |||||||

32 | WIGM | 1.20E-06 | 5.29 | 4.70E-05 | 4.69 | 1.21E-03 | 3.75 | 1.63E-02 | 2.46 | 8.89E-02 | |

WMIGM | 1.20E-06 | 5.29 | 5.73E-06 | 5.06 | 4.36E-05 | 6.75 | 5.27E-03 | 1.57 | 1.63E-02 | ||

64 | WIGM | 2.42E-08 | 5.64 | 1.20E-06 | 5.29 | 4.70E-05 | 4.69 | 1.21E-03 | 3.75 | 1.63E-02 | 2.45 |

WMIGM | 2.42E-08 | 5.64 | 1.25E-07 | 5.52 | 7.24E-07 | 5.91 | 4.09E-05 | 7.01 | 5.57E-03 | 1.55 | |

QBSM [ |
1.24E-07 | 1.95E-06 | 2.94E-05 | 3.40E-05 | 3.40E-05 | ||||||

128 | WIGM | 4.31E-10 | 5.81 | 2.42E-08 | 5.64 | 1.20E-06 | 5.29 | 4.70E-05 | 4.69 | 1.21E-03 | 3.75 |

WMIGM | 4.31E-10 | 5.81 | 2.31E-09 | 5.76 | 1.33E-08 | 5.77 | 6.51E-07 | 5.97 | 3.85E-05 | 7.18 | |

QBSM [ |
7.75E-09 | 3.99 | 1.24E-07 | 3.98 | 1.95E-06 | 3.91 | 2.57E-06 | 3.73 | 2.67E-06 | 3.67 | |

256 | WIGM | 2.25E-11 | 4.26 | 4.32E-10 | 5.81 | 2.42E-08 | 5.64 | 1.20E-06 | 5.29 | 4.70E-05 | 4.69 |

WMIGM | 2.25E-11 | 4.26 | 4.02E-11 | 5.85 | 2.32E-10 | 5.84 | 1.20E-08 | 5.77 | 5.96E-07 | 6.01 | |

QBSM [ |
4.85E-10 | 4.00 | 7.79E-09 | 3.99 | 1.24E-07 | 3.98 | 1.90E-07 | 3.76 | 2.01E-07 | 3.74 | |

512 | WIGM | - | - | - | 4.34E-10 | 5.80 | 2.42E-08 | 5.64 | 1.20E-06 | 5.29 | |

WMIGM | - | - | - | - | 1.36E-11 | 4.09 | 2.08E-10 | 5.85 | 1.09E-08 | 5.77 | |

QBSM [ |
3.03E-11 | 4.00 | 4.87E-10 | 4.00 | 7.79E-09 | 3.99 | 1.30E-08 | 3.87 | 1.59E-08 | 3.65 | |

1024 | WIGM | - | - | - | - | - | - | 4.39E-10 | 5.78 | 2.42E-08 | 5.64 |

WMIGM | - | - | - | - | - | - | 2.68E-11 | 2.96 | 1.84E-10 | 5.89 | |

QBSM [ |
1.76E-12 | 4.10 | 3.04E-11 | 4.00 | 4.87E-10 | 4.00 | 1.00E-09 | 3.69 | 1.39E-09 | 3.81 |

Finally, we consider the following homogeneous linear singularly perturbed boundary value problem [

with boundary conditions extracted from the exact solution as

The maximum absolute errors obtained by the QBSM with Shishkin mesh [

Method | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{max} |
R | _{max} |
R | _{max} |
R | _{max} |
R | _{max} |
R | ||

16 | WIGM | 8.40E-05 | 1.54E-03 | 1.76E-02 | 9.01E-02 | ||||||

WMIGM | 8.40E-05 | 2.54E-04 | 4.78E-03 | 1.63E-02 | |||||||

32 | WIGM | 2.29E-06 | 5.20 | 6.35E-05 | 4.60 | 1.37E-03 | 3.68 | 1.69E-02 | 2.41 | 8.94E-02 | |

WMIGM | 2.29E-06 | 5.20 | 7.89E-06 | 5.01 | 4.27E-05 | 6.81 | 5.29E-03 | 1.62 | 1.66E-02 | ||

64 | WIGM | 4.75E-08 | 5.59 | 1.68E-06 | 5.24 | 5.48E-05 | 4.64 | 1.29E-03 | 3.71 | 1.66E-02 | 2.43 |

WMIGM | 4.75E-08 | 5.59 | 1.76E-07 | 5.49 | 7.05E-07 | 5.92 | 4.01E-05 | 7.04 | 5.57E-03 | 1.57 | |

NOBS [ |
1.74E-09 | 7.86E-08 | 3.50E-06 | 8.26E-05 | 7.99E-04 | ||||||

QBSM [ |
1.79E-07 | 2.34E-06 | 3.21E-05 | 4.07E-04 | 4.92E-03 | ||||||

128 | WIGM | 8.56E-10 | 5.79 | 3.43E-08 | 5.61 | 1.42E-06 | 5.27 | 5.08E-05 | 4.67 | 1.25E-03 | 3.73 |

WMIGM | 8.56E-10 | 5.79 | 3.29E-09 | 5.74 | 1.30E-08 | 5.76 | 6.36E-07 | 5.98 | 3.80E-05 | 7.20 | |

NOBS [ |
2.79E-11 | 5.96 | 1.32E-09 | 5.89 | 6.85E-08 | 5.67 | 3.29E-06 | 4.65 | 7.82E-05 | 3.35 | |

QBSM [ |
1.12E-08 | 3.99 | 1.49E-07 | 3.98 | 2.14E-06 | 3.91 | 3.07E-05 | 3.73 | 3.98E-04 | 3.63 | |

256 | WIGM | 1.54E-11 | 5.80 | 6.12E-10 | 5.81 | 2.89E-08 | 5.62 | 1.31E-06 | 5.28 | 4.89E-05 | 4.68 |

WMIGM | 1.54E-11 | 5.80 | 5.60E-11 | 5.88 | 2.33E-10 | 5.80 | 1.17E-08 | 5.76 | 5.87E-07 | 6.02 | |

NOBS [ |
4.39E-13 | 5.99 | 2.12E-11 | 5.97 | 1.15E-09 | 5.90 | 6.39E-08 | 5.69 | 3.20E-06 | 4.61 | |

QBSM [ |
7.02E-10 | 4.00 | 9.37E-09 | 3.99 | 1.36E-07 | 3.98 | 2.05E-06 | 3.91 | 3.00E-05 | 3.73 | |

512 | WIGM | - | - | 2.74E-11 | 4.48 | 5.13E-10 | 5.82 | 2.65E-08 | 5.63 | 1.26E-06 | 5.28 |

WMIGM | - | - | - | - | 9.40E-12 | 4.63 | 2.12E-10 | 5.79 | 1.08E-08 | 5.77 | |

NOBS [ |
6.88E-15 | 6.00 | 3.33E-13 | 5.99 | 1.83E-11 | 5.97 | 1.07E-09 | 5.90 | 6.17E-08 | 5.70 | |

QBSM [ |
4.39E-11 | 4.00 | 5.86E-10 | 4.00 | 8.54E-09 | 3.99 | 1.30E-07 | 3.98 | 2.00E-06 | 3.91 | |

1024 | WIGM | - | - | - | - | 5.39E-11 | 3.25 | 4.67E-10 | 5.82 | 2.53E-08 | 5.63 |

WMIGM | - | - | - | - | - | - | 1.35E-11 | 3.97 | 1.98E-10 | 5.76 | |

NOBS [ |
1.08E-16 | 6.00 | 5.20E-15 | 6.00 | 2.88E-13 | 5.99 | 1.71E-11 | 5.96 | 1.03E-09 | 5.90 | |

QBSM [ |
2.77E-12 | 3.98 | 3.66E-11 | 4.00 | 5.35E-10 | 4.00 | 8.16E-09 | 3.99 | 1.27E-07 | 3.98 |

In this study, we extended the WMIGM to solve linear singularly perturbed boundary value problems with modified Shishkin nodes. The proposed wavelet scheme was verified by comparing the numerical solutions obtained via the QBSM with Shishkin mesh and the NOBS with Shishkin mesh. The numerical results confirm the theoretical analysis and demonstrate that WMIGM has several advantages over existing schemes:

The accuracy of the WMIGM is significantly better than that of the WIGM. The approximate solutions obtained by the WMIGM exhibit no obvious spurious oscillations near the boundary layer, even as the perturbation parameter approaches zero.

The WMIGM exhibits greater accuracy than existing schemes, including those of the QBSM and NOBS methods with Shishkin mesh.

The WMIGM demonstrates a six-order convergence rate and retains a stable convergence order better than that of the QBSM and NOBS methods with the Shishkin mesh.

These advantages indicate the potentially wide application of the WMIGM to simulating problems with local large gradients. Since the proposed WMIGM allows a very flexible nodal distribution, it can be extended to solve other problems with localized steep gradients, such as the steady-state convection diffusion problems, the steady-state heat transfer at high Péclet numbers, and the planar thin plate problems in solid mechanics. Additionally, the combination of the time integral format and the proposed method enables the solution of time dependent systems, including the Navier-Stokes equations with large Reynolds numbers and convective heat transfer problems with large Péclet numbers. Moreover, combining WMIGM with wavelet adaptive analysis holds great appeal as a potentially superior method for solving these intricate problems.

The authors sincerely thanks the reviewers and the editors of the journal for the great improvement of this paper.

This work was supported by the National Natural Science Foundation of China (No. 12172154), the 111 Project (No. B14044), the Natural Science Foundation of Gansu Province (No. 23JRRA1035), and the Natural Science Foundation of Anhui University of Finance and Economics (No. ACKYC20043).

The authors confirm contribution to the paper as follows: study conception and design: Jiaqun Wang, Xiaojing Liu; coding: Jiaqun Wang, Guanxu Pan; analysis and interpretation of results: Guanxu Pan, Xiaojing Liu; funding support: Jiaqun Wang, Youhe Zhou, Xiaojing Liu; draft manuscript preparation: Jiaqun Wang, Guanxu Pan; draft review: Youhe Zhou, Xiaojing Liu. All authors reviewed the results and approved the final version of the manuscript.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.