ajcm American Journal of Computational Mathematics 2161-1211 2161-1203 Scientific Research Publishing 10.4236/ajcm.2026.162008 ajcm-152181 Article Physics Mathematics Economic Cascadic Tensor Multigrid Method Based on Finite Element Discretization for 3D Partial Differential Equations 0009-0006-0445-137X Huang Jingyu 1 0000-0001-8166-0243 Li Chenliang 1 School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China

The authors declare no conflicts of interest regarding the publication of this paper.

 02 06 2026 06 2026 16 02 143 150 03 05 2026 26 06 2026 29 06 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/ajcm.2026.162008

In this paper, based on Tucker product tensor form, we present an economic cascadic tensor multigrid method(ECTMG) for the 3D partial differential equations discretized by the finite element method. Meanwhile, we provide the convergence analysis of the new method. Finally, we verify the effectiveness of the new method through some numerical examples.

 Partial Differential Equations Economic Cascadic Tensor Multigrid Method Tucker Product Finite Element Method 1. Introduction

The finite element method is widely applied in fields such as industrial equipment [1], biomedicine [2][3], and new energy catalysis [4]. Partial differential equations are discretized into linear equation systems (1) through the finite element method [5].

A x = b ,

A = ( ∑ k = 1 N   I J N ⊗ ⋯ ⊗ I J k + 1 ⊗ A ( k ) ⊗ I J k − 1 ⊗ ⋯ ⊗ I J 1 ) , A ( k ) ∈ ℝ J k × J k ( k = 1 , ⋯ , N ) , A ( k ) ⊗ I J k − 1 represent as follows

A ( k ) ⊗ I J k − 1 = [ a 11 I J k − 1 ⋯ a 1 J 2 I J k − 1 ⋮ ⋱ ⋮ a 1 J 1 I J k − 1 ⋯ a J 1 J 2 I J k − 1 ] .

Chen and Lu [6] equivalently transform the linear system (1) into the following Sylvester tensor equation

X × 1 A ( 1 ) + X × 2 A ( 2 ) + ⋯ + X × N A ( N ) = D ,

with A ( k ) ∈ ℝ J k × J k ( k = 1 , ⋯ , N ) and D ∈ ℝ J 1 × ⋯ × J N , X ∈ ℝ J 1 × J 2 × ⋯ × J N . The mode-k product of X and A is denoted by X × k A ( k ) , the result is of size J 1 × ⋯ × J k − 1 × J × J k + 1 × ⋯ × J N , and its entries are defined by

( X × k A ( k ) ) i 1 ⋯ i k − 1 j i k + 1 ⋯ i N = ∑ i k = 1 J k   x i 1 ⋯ i k − 1 i k i k + 1 ⋯ i N a j i k ( k ) .

Compared with Equation (1), Equation (2) reduces the storage space required for the solution process, thereby improving the solution efficiency. Grasedyck et al. [5] utilized the finite element method and expressed A ( i ) in Equation (2) as follows

A ( i ) = ( M ( i ) ) − 1 A ˜ ( i ) .

Here, M ( i ) represents the mass matrix, and A ˜ ( i ) is the coefficient matrix obtained by the finite element method.

The structure of this paper is as follows. In Section 2, we discretize the high-dimensional partial differential equation by the finite element method, and give the convergence analysis of the new method. Section 3 shows and discusses the numerical results. Finally, in Section 4, we conclude this article with some content.

 2. Main Result 2.1. Discretization of 3D PDE by Cubic Element

Let Ω ⊂ ℝ 3 be a bounded Lipschitz domain with boundary ∂ Ω = Γ D ∪ Γ N , Γ D ∩ Γ N = ∅ , and meas ( Γ D ) > 0 . Consider the following elliptic boundary value problem:

{ − ∇ ⋅ ( A ∇ u ) + b ⋅ ∇ u + c u = f in   Ω , u = 0 on   Γ D , ( A ∇ u ) ⋅ n = g on   Γ N ,

where

A ( x ) ∈ ℝ 9 × 9 is symmetric and positive definite: there exist constants 0 < α ≤ β < ∞ such that

α ‖ ξ ‖ 2 ≤ ξ T A ( x ) ξ ≤ β ‖ ξ ‖ 2       ∀ ξ ∈ ℝ 3 ,     a . e .   x ∈ Ω ;

b ( x ) ∈ ℝ 3 is a bounded vector field; c ( x ) ≥ 0 is a bounded scalar; f ∈ L 2 ( Ω ) , g ∈ L 2 ( Γ N ) are given data; n is the outward unit normal on Γ N .

Define the function space V = { v ∈ H 1 ( Ω ) : v | Γ D = 0 } . Multiplying the PDE by a test function v ∈ V and integrating by parts yields the weak problem: find u ∈ V such that

a ( u , v ) = ℓ ( v )       ∀ v ∈ V ,

with

{ a ( u , v ) = ∫ Ω ( ( A ∇ u ) ⋅ ∇ v + ( b ⋅ ∇ u ) v + c u v ) d x , ℓ ( v ) = ∫ Ω f v d x + ∫ Γ N g v d s .

Then its variational form is as follows

− ∭ V [ ∇ ⋅ ( A ∇ u ) ] v d V + ∭ V ( b ⋅ ∇ u ) v d V + ∭ V c u v d V = ∭ V f v d V .

Thus, the complete weak form is obtained

∭ V ( A ∇ u ⋅ ∇ v + ( b ⋅ ∇ u ) v + c u v ) d V − ∬ ∂ V v ∂ u ∂ n A d S = ∭ V f v d V .

Let

N x ( l ) , N y ( l ) , N z ( l ) = 2 l + 1 , h = 1 N i ( l ) + 1 ( i = x , y , z )

it can be obtained that

Δ x = x L − x 0 N x ( l ) , Δ y = y L − y 0 N y a ( l ) , Δ z = z L − z 0 N z ( l ) .

Figure 1 presents the cube element.

Figure 1.Cube element.

We write the following basic functions,

L ^ 1 = ξ ( 1 − η ) ( 1 − γ ) , L ^ 2 = ξ η ( 1 − γ ) , L ^ 3 = ξ ( 1 − η ) γ , L ^ 4 = ξ η γ , L ^ 5 = ( 1 − ξ ) ( 1 − η ) ( 1 − γ ) , L ^ 6 = ( 1 − ξ ) η ( 1 − γ ) , L ^ 7 = ( 1 − ξ ) ( 1 − η ) γ , L ^ 8 = ( 1 − ξ ) η γ .

And the coordinates are mapped as

x = ∑ i = 1 8   x i L ^ i ( ξ , η , γ ) , y = ∑ i = 1 8   y i L ^ i ( ξ , η , γ ) , z = ∑ i = 1 8   z i L ^ i ( ξ , η , γ ) .

Furthermore, it can be concluded that the Jacobian matrix J is

J = | ∂ x ∂ ξ ∂ y ∂ ξ ∂ z ∂ ξ ∂ x ∂ η ∂ y ∂ η ∂ z ∂ η ∂ x ∂ γ ∂ y ∂ γ ∂ z ∂ γ | .

Therefore,

K i j = ∭ e 1 ∇ L i ∇ L j ∇ L k d x d y d z = ∭ e ^ 1 ∇ L ^ i ∇ L ^ j ∇ L ^ k | J | d ξ d η d γ .

After assembling the stiffness matrix of the assembly unit, the linear system (1) can be obtained,

A l x l = b l .

We have converted it equivalently to

X × 1 A l ( 1 ) + X × 2 A l ( 2 ) + X × 3 A l ( 3 ) = D l ,

where, A l ( i ) ∈ ℝ J k × J k ( k = 1 , 2 , 3 ) and D l is the right-hand term, the structure of coefficient matrix A l ( i ) is as follows

A l ( i ) = ( M l ( i ) ) − 1 A ˜ l ( i ) ,

Here, M l ( i ) represents the mass matrix, and A ˜ l ( i ) is the coefficient matrix obtained by the finite element method.

 2.2. Inverse of Mass Matrix <inline-formula> <mml:math> <mml:mstyle mathvariant="bold" mathsize="normal"> <mml:mi>M</mml:mi> </mml:mstyle> </mml:math> </inline-formula>

According to [7], we can use the Stenger quadrature formula to approximate the inverse of the mass matrix M .

Let M ∈ ℝ n × n be a symmetric positive definite matrix, with all eigenvalues λ > 0 . Consider the integral

∫ 0 ∞ e t M d t .

For each eigenvalue λ , the integral is

∫ 0 ∞ e − λ t d t .

Due to the relationship between the matrix exponential and the eigenvalues, we have

∫ 0 ∞ e − t M d t = M − 1 .

Let p ∈ ℕ , Step length is h st = π 2 / p , the integration nodes and weights are as follows

t j = ln ( e j h st + 1 + e 2 j h st ) ,     j = − p , ⋯ , p . w j = h st 1 + e − 2 j h st .

Then the integral is approximately equal to

∫ 0 ∞ e − t λ d t ≈ ∑ j = − p p   w j e − t j λ . 2.3. ECTMG Algorithm

Combining Sections 2.1 and 2.2, we present the following iterative method (Algorithm 1).

Algorithm 1. The calculation of A ( i )

Similar to [7], we employ the following algorithm for the solution (Algorithm 2).

Algorithm 2.ECTMG algorithm [7].

With

is prologation [8], T l m l is the smoother. And m l represents the number of iterations, we similar to [9] provides the following selection rules:

d = 2

1) If l > L 0 , then m l = [ m L β L − l ] .

2) If l ≤ L 0 , m l = [ m * 1 2 ( L − ( 2 − ε 0 ) l ) κ l ] .

d ≥ 3

1) If l > L 0 , then m l = [ m L β L − l ] .

2) If l ≤ L 0 , there are two cases:

a) If ( 2 − ε 0 ) L 0 ≤ L , then m l = [ m * 1 2 ( L − ( 2 − ε 0 ) l ) κ l ] .

b) If ( 2 − ε 0 ) L 0 > L , there exists a positive integer L ′ < L 0 such that ( 2 − ε 0 ) L ′ ≤ L , for all l ≤ L ′ , choose m l = [ m * ( L − ( 2 − ε 0 ) l ) κ l ] .

Remark Let κ l = h l − 2 , and ε 0 be a positive constant in the interval [0, 1].

 3. Numerical Example

In this section, we verify the effectiveness of the ECTMG algorithm through some numerical examples. All tests will be done with configuration: 11th Gen Intel(R) Core(TM) i5-11400H @ 2.70 GHz 2.69 GHz. Let CPU(s) represent the iteration time. ε 0 = 0.1 .

We define the error between the exact solution X * and the numerical solution X k as follows

E ( X * − X k ) = max i 1 , i 2 , ⋯ , i N | x i 1 , i 2 , ⋯ , i N * − x i 1 , i 2 , ⋯ , i N k | ,

Obviously, we have E ( X * − X k ) = ‖ v e c ( X * − X k ) ‖ ∞ . In the table of numerical examples in the following text, the symbol † indicates insufficient memory during the algorithm’s execution.

Example1. ([8]) ConsiderthefollowingPoissonequation

− △ u = f in V = [ 0 , 1 ] 3 u = 0 on ∂ V .

Taking the partition step length as h = 1 n + 1 , using the hexahedral element, we can obtain the Sylvester tensor Equation (1), with A ( k ) in Equation (2) as follows

A ( k ) = ( M ( k ) ) − 1 A ˜ ( k ) ,

with A ˜ ( k ) as follows

A ˜ ( k ) = 2 9 [ 4 − 2 0 − 2 4 − 2 0 − 2 4 ] .

and M ( k ) as follows

M ( k ) = 1 4 [ 4 1 0 1 4 1 0 1 4 ] .

Set V l = 2 / ∑ k = 1 3 ( α l k β l k ) , where α l k and β l k are the maximum and minimum eigenvalues of matrix A l ( k ) , respectively. Select the right-hand side tensor ℬ such that the solution to the corresponding 3-order Poisson equation is

u = 1 3 ⋅ ∏ k = 1 3 ( x k − x k 2 )

lists the numerical results of BiCG_BTF and ECTMG(BiCG_BTF) algorithms.

Table 1.Numerical results for n = 255 × 255 × 255 .

Algorithm CPU (s) E ( X * − X L ) IT
BiCG_BTF 663.32 2.20 × 10 − 7 994
ECTMG (BiCG_BTF) 142.81 4.72 × 10 − 7 (3, 205, 256, 1024, 4096, 512, 64)

Remark: In the BiCG_BTF algorithm, IT represents the total number of iterations required for the entire algorithm process; in the ECTMG algorithm, it is the number of iterations m l for the corresponding level.

Table 2.Numerical results for n = 511 × 511 × 511 .

Algorithm CPU (s) E ( X * − X L ) IT
BiCG_BTF 14076.29 4.96 × 10 −7 1958
ECTMG (BiCG_BTF) 790.07 5.78 × 10 −7 (3, 269, 256, 1024, 15,360, 1920, 240, 30)

As can be seen from Table 1 and Table 2, through the linear system (2) discretized by the finite element method, the ECTMG algorithm can still efficiently solve the problem and maintain the same accuracy as BiCG_BTF, which verifies the feasibility of the finite element discretized equation for this solution framework.

 4. Conclusion

Based on the finite element method, we discretize the three-dimensional partial differential equations and solves it using the economic cascadic tensor multigrid method. The numerical results show that the new method is effective.

 Acknowledgements

We acknowledge the financial support from the Natural Science Foundation of China under Grant 12161027 and the Science and Technology Project of Guangxi (Guike AD25069086).

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