2. L-ORBFs with Augmented Polynomial Terms
In this section, we present the proposed numerical approach, L-ORBFs with augmented polynomials, in the localized collocation process. For simplicity, we consider following elliptic PDEs with a Dirichlet boundary condition:
where
are constant or variable coefficients;
,
is the analytical solution;
are partial derivatives of
v with respect to
, respectively;
and
are, respectively, the interior and boundary of the computational domain; and
and
are the known functions.
Consider
computational nodes with
interior nodes in
and
boundary nodes on
such that
. We choose one of the ORBFs [
4] defined by
where
is the Euclidean norm,
is the shape parameter,
signifies the
J Bessel function of the first kind of order
, and
is called the ORBF of order
d. To solve the elliptic PDEs (
1) and (2) using the proposed scheme, we require the particular solutions for
, derived in [
8], as follows:
For
where
is the double factorial defined as
and
is the cosine integral function.
For
we use
Details on deriving the particular solutions using ORBFs can be found in [
8].
Next, we choose
polynomials
up to a degree of
m as follows:
Note that For example, if we choose polynomials up to a degree of three, then we augment 10 polynomial terms .
Now, we choose
nearest nodes
at each node
to form a corresponding local domain
. Then, at each node
we approximate
by
where
and
are the additional polynomial terms. Because of the addition of
terms in (
4), we need to add the following
constraints in the system of equations:
For
(
4) and (
5) imply a matrix equation
where
and are zero matrices of sizes and , respectively.
To solve (
1) and (2), we assume that for each
, (
4) satisfies (
1) and (2). This implies that for
Using the linearity of a differential operator, we obtain
where
. This implies that
where
and
is obtained by inserting
zeros in
based on mapping between
and
Thus,
For
in general, if we have any boundary operator
, we obtain
where
and
is obtained by inserting
zeros in
based on mapping between
and
For the identity boundary operator in (2), we can create the vector whose entry is 1 and other entries are 0.
The approximate solution
of (
1) and (2) at each of the given computational nodes is obtained by solving Equations (
8) and (
9).
3. Numerical Results
We perform three numerical experiments to verify our proposed numerical approach. The R2021a built-in functions KDTreeSearcher and knnsearch are used to find the nearest neighborhood nodes on the local computational domain . These built-in functions are available in the statistics and machine learning toolbox.
The numerical accuracy is measured by the root mean square error (RMSE) and
error, which are defined as
and
Unless otherwise specified, on most of the numerical experiments, we choose the first order of ORBFs
, and the shape parameter
is chosen by a heuristic approach. Each local domain
contains
nearest neighborhood nodes, and the augmented polynomial terms are of a degree up to
. Along with the unit square domain
, we have used some of the irregular domains bounded by the following parametric curves:
where
amoeba-shaped curve (
Figure 1a);
Cassini-shaped curve (
Figure 1b);
star-shaped curve (
Figure 1c);
gear curve (
Figure 1d) with 15 cogs.
Another type of gear curve (
Figure 1e) with six cogs used in the experiment is defined as
where
Figure 1.
Irregularly shaped curves used in the experiments to bound the computational domain.
Figure 1.
Irregularly shaped curves used in the experiments to bound the computational domain.
Example 1. We consider the elliptic PDEs (1) and (2) with ; and are defined based on the analytical solution given belowwhere are constants. First, we investigate the numerical accuracy of the L-ORBFs and the L-ORBFs with augmented polynomial terms in the localized collocation process for solving the Poisson’s equation given in this example. For this numerical experiment, a family of exponential-type analytical solutions is chosen. In
Table 1, we present the RMSE of the L-ORBFs and the proposed approach with different values of
and
in the analytical solution
. A unit square domain is considered with
interior nodes and
boundary nodes. Generally, the L-ORBFs is suitable for approximating the functions that are oscillatory in nature, but the result shows that the proposed approach not only performs better than the L-ORBFs but also solves PDEs with high accuracy when the analytical solutions are exponential in nature.
Without loss of generality, we choose
. In
Figure 2, we observe the highly accurate approximate solution
of the analytical solution
over a unit square domain. The error distribution plot in
Figure 2c shows that the proposed approach is highly accurate everywhere in the domain except for a small region with
× 10
.
As meshless methods are renowned for dealing with irregular domains, we have compared our proposed approach with the L-ORBFs for solving the given Poisson’s equation defined over the domain bounded by irregular curves.
Figure 3a depicts the profile of the analytical solution
over the computational domain bounded by the gear-shaped curve with six cogs. We choose
interior nodes and
boundary nodes over the domain. The RMSE vs. the error profile of the shape parameter
depicted in
Figure 3b clearly shows that the proposed approach produces better numerical accuracy (RMSE =
× 10
at
) than the L-ORBFs (RMSE =
× 10
at
).
Figure 4 depicts the numerical results obtained over a computational domain (
,
) bounded by the Cassini-shaped curve. In
Figure 4a, we observe that the proposed approach with the fourth degree augmented polynomial terms has performed better than the L-ORBFs in terms of the numerical accuracy. A similar phenomenon can be observed in
Table 2, which provides the comparison of the RMSEs between the L-ORBFs and the proposed approach for the domains bounded by different irregular curves.
Figure 5 portrays the profile of the RMSE vs. the shape parameter for the star-shaped and amoeba-shaped domains. In both cases, our proposed approach undoubtedly surpassed the accuracy of the L-ORBFs.
Figure 4b shows the profile of the RMSE vs. the degree
m of the augmented polynomials. We clearly notice from
Figure 4b that when we increase the degree
m of the augmented polynomials, we obtain an increasingly accurate solution. During our experiment, we also observed that when the total number of polynomial terms is closer to or surpasses the
nearest neighborhood nodes, then the method is unstable and ceases to produce results. Therefore, we have chosen low-degree polynomials in our proposed approach. If more polynomial terms are to be used, then we need to increase the size of the local computational domain, which provides highly accurate solutions; however, this will be computationally more expensive.
Next, we solve Example 1 with a large number of computational nodes. A unit square domain is considered using
as the interior and 2080 as the boundary nodes.
Table 3 presents the numerical results for the varied number
of nearest neighborhood nodes on the local domains. We clearly observed that the numerical accuracy is consistently around 1 × 10
for a varied number of local computational nodes.
In
Table 4, we investigate the numerical accuracy of the proposed approach by varying the highest degree
m of the augmented polynomial terms. We observed that the increased
m produces a more accurate solution while increasing the computational cost. We note that the RMSE using the L-ORBFs is equal to
× 10
in less than 80 s, but, using the proposed approach, the accuracy increases to
× 10
in 133 s.
Example 2. Next, we choose a modified Helmholtz problem with , , and in (1) and (2). The functions and are considered based on the following analytical solution A star-shaped domain is considered throughout this example to observe the accuracy of the proposed method. The profile of the analytical solution over the star-shaped domain is presented in
Figure 6.
Table 5 displays the numerical results for various sets of large-scale computational nodes. The size of the local computational domain is chosen to be
and the wavelength is
in this experiment. The RMSE and
error for different degrees (
) of the augmented polynomials are presented in this table. It is clearly observable that the accuracy of the proposed method significantly improves when we increase the degree of the augmented polynomials. However, the computational cost becomes higher with the increase in the degree of the augmented polynomials. A descending numerical accuracy is obtained by adding low-degree polynomials to the ORBFs, and we are able to validate our claim that the polynomial augmentations process on the L-ORBFs clearly outperforms the accuracy of the L-ORBFs.
Furthermore,
Table 6 presents the numerical accuracy of our proposed scheme for different values of the wavelength
in the modified Helmholz problem. The interior and boundary nodes on the star-shaped domain are taken as 51,278 and 1000, respectively. The number of local computational nodes is
, and the highest degree of the augmented polynomials is
for this numerical experiment. It is clear from the results that the numerical accuracy improves as we increase the wavelength in the problem. Therefore, we conclude that our method is very promising and highly accurate.
Example 3. Finally, we choose the elliptic PDEs with in (1) and (2); and are considered based on the following analytical solution First, we choose the computational domain bounded by the amoeba-shaped curve (
Figure 1a). The profile of the analytical solution
over an amoeba-shaped domain is depicted in
Figure 7a.
In this experiment, we have chosen
interior nodes and
boundary nodes in the amoeba-shaped domain. In
Table 7,
Table 8 and
Table 9, we present the RMSE and
error for solving elliptic PDEs with variable coefficients.
Table 7 presents the numerical results of the L-ORBFs for various sizes of the local computational domain
, whereas
Table 8 and
Table 9 present the numerical results of the L-ORBFs with augmented polynomials.
Comparing numerical results from
Table 7,
Table 8 and
Table 9, we observe significant improvement in the numerical accuracy while employing the proposed scheme. When the number of local computational nodes increases in the L-ORBFs, it provides consistent accuracy around 1 × 10
(
Table 7). On the other hand, when
are chosen, the proposed scheme reaches accuracy around 1 × 10
(
Table 9). Although the results in these tables are obtained for a specific
chosen by a heuristic approach, we can observe from
Figure 7b that the proposed scheme is more highly accurate than the L-ORBFs.
Table 8 depicts the numerical results using the proposed scheme corresponding to the local computational nodes
and 35. Various degrees of augmented polynomial terms (
) are chosen. As expected, adding more polynomial terms not only increases accuracy but also increases computational cost so it is not a wise idea to add a lot of polynomial terms in the localized collocation process. This indicates that if we would like to improve the accuracy, it is better to add only a few more polynomial terms. In the table, with
, we only need to add polynomial terms up to
to achieve accuracy up to 1 × 10
, which means by only adding
more polynomial terms on the localized collocation process, we can achieve a highly accurate solution, and it only takes a time of 40 s.
Next, we choose the computational domain bounded by the gear curve with 15 cogs (
Figure 1d). The surface plot of the analytical solutions
over a gear-curve domain is depicted in
Figure 8.
Let us implement our numerical scheme using
interior nodes, and 6000 boundary nodes in the gear-curve domain. For a large number of computational nodes, we believe that it makes sense to take more local computational nodes to capture the local calculus of the domain.
Table 10 presents the numerical results obtained by the L-ORBFs, and
Table 11 shows the results of the proposed scheme. In this experiment, we have only included the polynomial terms up to the degree
. We notice that even with the large number of computational nodes, similar conclusions are perceived. The proposed scheme is more highly accurate than the L-ORBFs. Comparing the results presented in
Table 10 and
Table 11 for the different number of local computational nodes, the L-ORBFs only produce accuracy around 1 × 10
while our proposed approach reaches accuracy closer to 1 × 10
. This clearly validates our claim.
Finally, we would like to comment on the varying order
d of ORBFs. Earlier research shows that if we choose an optimal shape parameter for each order
d, it produces a reasonably accurate solution in the L-ORBFs. Without loss of generality, throughout the paper, we have fixed the order of ORBFs to
. Ultimately, we want to verify that this is not only true in the L-ORBFs but also in this proposed scheme.
Table 12 presents the RMSE and
error for a various order
d of ORBFs on a gear-curve domain with
and
. For this experiment, we have chosen
and
. We observed that varying the order
d of ORBFs does not change the final outcome of the solution significantly, but it definitely changes the computational time. For
, to achieve the RMSE =
× 10
and
× 10
, it takes 265 s. On the other hand, similar accuracy can be achieved by choosing
within 195 s. Note that these computational costs vary by order due to the evaluation cost of ORBFs and their corresponding particular solutions.