A parallel iterative solution method and system for electromagnetic finite element equations

Abstract

The present invention provides a parallel iterative solution method and system for electromagnetic finite element equations, which relates to the field of electromagnetic finite element technology, including: constructing an electromagnetic finite element equation to be solved; dividing the electromagnetic finite element equation into a plurality of computing subtasks, and allocating the sub-matrix blocks and sub-right-hand-end term vectors of the subtasks to each process in a supercomputer; using the initialized Householder‑GMRES algorithm, each process performs the subtask calculation in parallel to obtain a local optimal solution of each process; summarizing the local optimal solutions of each process to obtain the optimal solution of the electromagnetic finite element equations; the present invention combines electromagnetic finite elements, the Householder‑GMRES algorithm, and parallel computing, and uses parallel optimization technology to improve the efficiency of solving electromagnetic finite element linear equations.

Description

Technical Field

[0001] The present disclosure relates to the field of electromagnetic finite element technology, and in particular to a parallel iterative solution method and system for electromagnetic finite element equations. Background Art

[0002] With the rise of a new round of industrial revolution such as smart manufacturing and smart cities and the rapid development of modern electronic communication technology, electromagnetic finite element technology has been widely used in various industries. However, the numerical calculation of complex electromagnetic finite element problems requires a large number of grids, and the solution process will consume a lot of time, which increases the efficiency requirements of electromagnetic finite element analysis. For complex electromagnetic problems, it is an urgent need to study efficient numerical calculation methods in the field of electromagnetic finite element technology.

[0003] The finite element method is a numerical calculation method for solving approximate solutions to boundary value problems of partial differential equations. With its powerful computing power, it is widely used in the field of electromagnetic finite element. Solving the linear equations in the electromagnetic field distribution is the most time-consuming part of the finite element calculation. In order to improve the efficiency of electromagnetic finite elements, it is necessary to optimize the solution of the equations in the finite element numerical simulation.

[0004] GMRES (Generalized Minimal Residual) is a common method for solving sparse linear equations. It approaches the solution of the equation by obtaining the vector that minimizes the residual in the Krylov subspace. Householder transformation is an efficient linear algebra method that reduces the complexity of the problem through matrix transformation, which is very helpful for solving linear systems. Introducing Householder transformation in the GMRES algorithm can effectively improve the stability and convergence speed of the algorithm. When the matrix size of the linear equation system increases and the number of iterations increases, the amount of calculation and storage required by the Householder-GMRES algorithm will also increase, and the solution work will be difficult to carry out. Designing a parallel solution algorithm with the help of supercomputers is an effective way to solve this problem. However, there is currently a lack of relevant optimization research on the parallel Householder-GMRES algorithm, which makes it difficult for the Householder-GMRES algorithm to give full play to its advantages such as high calculation accuracy and strong stability; therefore, the existing electromagnetic finite element equation solution method lacks an efficient parallel implementation method, which limits the efficiency of electromagnetic finite element analysis. Summary of the invention

[0005] In order to solve the above problems, the present invention proposes a parallel iterative solution method and system for electromagnetic finite element equations, which combines electromagnetic finite element, Householder-GMRES algorithm and parallel computing, and uses parallel optimization technology to improve the efficiency of solving electromagnetic finite element linear equations.

[0006] According to some embodiments, the present disclosure adopts the following technical solutions:

[0007] A parallel iterative solution method for electromagnetic finite element equations, comprising:

[0008] Construct the electromagnetic finite element equations to be solved , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, and free space wave number; represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential;

[0009] Divide the electromagnetic finite element equations into multiple computational subtasks and block the submatrices of the subtasks and the right-hand side vector Assigned to each process in the supercomputer, where Number the process;

[0010] Using the initialized Householder-GMRES algorithm, each process performs subtask calculations in parallel to obtain the local optimal solution of each process. ;

[0011] Local optimal solution for each process Summarize and obtain the optimal solution of the electromagnetic finite element equations ;

[0012] The calculation of the subtask is an iterative calculation of Householder orthogonalization, Givens transformation and trigonometric equation solution in each process in turn until the condition for stopping the iteration is met and the local optimal solution is obtained. , during the iterative calculation process, the respective residuals , non-zero vector , unit vector , basis vector , a set of standard orthogonal basis , transformation matrix , Shanghai Senburger Array , local optimal solution A global inter-process communication protocol.

[0013] According to some embodiments, the present disclosure adopts the following technical solutions:

[0014] A parallel iterative solution system for electromagnetic finite element equations, comprising:

[0015] A building block configured to: build the electromagnetic finite element equations to be solved , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, and free space wave number; represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential;

[0016] The partitioning module is configured to: partition the electromagnetic finite element equations into multiple computational subtasks, and and the right-hand side vector Assigned to each process in the supercomputer, where Number the process;

[0017] The calculation module is configured as follows: using the initialized Householder-GMRES algorithm, each process performs subtask calculations in parallel to obtain the local optimal solution of each process ;

[0018] The aggregation module is configured to: Summarize and obtain the optimal solution of the electromagnetic finite element equations ;

[0019] The calculation of the subtask is an iterative calculation of Householder orthogonalization, Givens transformation and trigonometric equation solution in each process in turn until the condition for stopping the iteration is met and the local optimal solution is obtained. , during the iterative calculation process, the respective residuals , non-zero vector , unit vector , basis vector , a set of standard orthogonal basis , transformation matrix , Shanghai Senburger Array , local optimal solution A global inter-process communication protocol.

[0020] According to some embodiments, the present disclosure adopts the following technical solutions:

[0021] A computer program product comprises a computer program, wherein when the computer program is executed by a processor, the parallel iterative solution method for electromagnetic finite element equations is implemented.

[0022] According to some embodiments, the present disclosure adopts the following technical solutions:

[0023] A non-transitory computer-readable storage medium is used to store computer instructions. When the computer instructions are executed by a processor, a parallel iterative solution method for electromagnetic finite element equations is implemented.

[0024] According to some embodiments, the present disclosure adopts the following technical solutions:

[0025] An electronic device comprises: a processor, a memory and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory so that the electronic device executes the parallel iterative solution method for electromagnetic finite element equations.

[0026] Compared with the prior art, the present invention has the following beneficial effects:

[0027] The present invention discloses a parallel iterative solution method for electromagnetic finite element equations, which uses a supercomputer to improve the efficiency of solving electromagnetic finite element linear equations. Specifically,

[0028] (1) The present invention makes full use of the advantages of multiple processes and divides the task of solving the electromagnetic finite element linear equations into multiple computing tasks according to the number of equations. Each process performs computational calculations and solves in parallel to achieve load balancing as much as possible.

[0029] (2) The present invention adopts different parallel optimization methods and inter-process communication methods in modules such as Householder orthogonalization, Givens transformation, and trigonometric equation solving, thereby improving memory access efficiency and shortening calculation time.

[0030] (3) The maximum acceleration ratio in the equation solution stage of the present invention reaches 26.02, which improves the electromagnetic finite element solution speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The accompanying drawings constituting a part of the present disclosure are used to provide a further understanding of the present disclosure. The illustrative embodiments of the present disclosure and their descriptions are used to explain the present disclosure and do not constitute an improper limitation on the present disclosure.

[0032] Figure 1 This is a flow chart of the method of Example 1.

[0033] Figure 2 This is a schematic diagram of the global communication protocol between processes in Example 1.

[0034] Figure 3 This is a time comparison chart before and after the Householder-GMRES parallel optimization of Example 1. DETAILED DESCRIPTION

[0035] The present disclosure is further described below in conjunction with the accompanying drawings and embodiments.

[0036] It should be noted that the following detailed descriptions are exemplary and are intended to provide further explanation of the present disclosure. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present disclosure belongs.

[0037] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present disclosure. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0038] Example 1

[0039] In one embodiment of the present disclosure, a parallel iterative solution method for an electromagnetic finite element equation group is provided, comprising:

[0040] Step S1: Construct the electromagnetic finite element equations to be solved , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, and free space wave number; x represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential;

[0041] Step S2: Divide the electromagnetic finite element equations into multiple computational subtasks and convert the submatrix blocks of the subtasks into and the right-hand side vector Assigned to each process in the supercomputer, where Number the process;

[0042] Step S3: Using the initialized Householder-GMRES algorithm, each process performs subtask calculations in parallel to obtain the local optimal solution of each process. ;

[0043] Step S4: Local optimal solution for each process Summarize and obtain the optimal solution of the electromagnetic finite element equations ;

[0044] The calculation of the subtask is an iterative calculation of Householder orthogonalization, Givens transformation and trigonometric equation solution in each process in turn until the condition for stopping the iteration is met and the local optimal solution is obtained. , during the iterative calculation process, the respective residuals , non-zero vector , unit vector , basis vector , a set of standard orthogonal basis , transformation matrix , Shanghai Senburger Array , local optimal solution A global inter-process communication protocol.

[0045] As an embodiment, the present invention discloses a parallel iterative solution method for electromagnetic finite element equations. It aims at the optimization needs of the Householder-GMRES algorithm. By introducing the Householder transformation into the GMRES algorithm, the electromagnetic finite element linear equations are efficiently solved in parallel to obtain the electromagnetic field distribution, which effectively improves the stability and convergence speed of the algorithm and has reference significance for electromagnetic finite elements. The specific implementation process is as follows: Figure 1 As shown:

[0046] Step 1: Construct the electromagnetic finite element equations to be solved

[0047] In order to obtain the electromagnetic field distribution, the Maxwell equations describing the interaction between the electric field and the magnetic field are added with some boundary conditions, and then the finite element method is used to construct a mathematical model. By discretizing space and time, the model is converted into a linear equation system, namely the electromagnetic finite element equation system to be solved. This discretization of the Maxwell equations converts the continuous electromagnetic field problem into a discrete mathematical problem, solves the physical quantity of the electromagnetic field, the electric field intensity, and then obtains the electromagnetic field distribution.

[0048] The electromagnetic finite element equations to be solved are expressed as , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, free space wave number, etc.; x represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential.

[0049] Step 2: Initialize the MPI environment and divide the electromagnetic finite element equations into tasks

[0050] Initialize the MPI environment and start multiple processes. Since the coefficient matrix A is sparse, it is compressed and stored in CSR format to save storage space. The task division method is to divide it into multiple computing subtasks evenly according to the number of equations in the equation group, and each process undertakes its own computing subtask; if it cannot be completely evenly divided, the last process will undertake the remaining subtasks.

[0051] If the number of available processes in the supercomputer is K, and the linear equation system contains N equations, the first K-1 processes are responsible for calculating equations, the Kth process is responsible for equations, and n is the number of equations that a single process is responsible for after task division.

[0052] The data storage method is that each process is divided according to the task and stores its own sub-matrix block separately. , right hand side vector , other sub-vectors and sub-matrices used in the following steps, where is the process number, .

[0053] Step 3: Set the initial parameters of Householder-GMRES

[0054] set up m dimensional Krylov subspace ,in, is the residual amount, m is the Krylov subspace dimension; the initial solution Set to 0; set the maximum number of iterations of the outer loop IterMax to prevent infinite iterations when there is no convergence; in addition to the maximum number of iterations, the condition of the outer loop also sets the residual accuracy requirement γ.

[0055] Step 4: Parallel Householder Orthogonalization

[0056] Sub-matrix block based and the right-hand side vector , each process is only responsible for calculating its own residual , unit vector , a set of standard orthogonal basis , transformation matrix And generate the Heisenberg array , non-zero vector ; The linear equations have equations, each process is responsible for indivual, Represents the first element values, Represents the number of iterations of the inner loop (i.e., the inner loop included in the Householder orthogonalization process), and the value range is , Represents the number of iterations of the outer loop, and the value range is Or if the residual accuracy requirement γ is met, the orthogonal parallel computing process is as follows:

[0057] First, calculate the residual , It is the initial setting.

[0058] Then, due to the subsequent update , so Assign to a non-zero vector ,Right now .

[0059] Next, the Householder transformation:

[0060] (1) Because it is used 2-norm of a vector , and each process only stores its own , each process first calculates its own Then use the MPI group communication function MPI_Allreduce to reduce and sum the processes. Each process gets the summed value and then takes the square root to get , and at the same time Assigning values ​​to variables .

[0061] The group communication function MPI_Allreduce here performs inter-process reduction and summation, which is implemented as follows Figure 2 The inter-process global communication protocol shown is equivalent to each process in the communication domain acting as a root process, performing a reduction and summation operation on the variable S. The result of the reduction is not only owned by a certain process, but by all processes.

[0062] (2) Unit vector The calculation is slightly more complicated because each process performs a The operations performed are different, and each stores The data is also different. According to the following formula (1), we first determine whether each process calculates a unit vector. Which part, and then the respective unit vector Calculation:

[0063] (1)

[0064] (2)

[0065] Where sign is the sign function, and its value is as shown in formula (2). is a unit vector, the first element is 1, and the others are 0; When doing so, the MPI group communication function MPI_Allreduce is also used to reduce and sum the numbers and then take the square root.

[0066] (3) Transformation matrix The parallel computing depends on the formula (3) , since each process previously only stored unit vectors A portion of the element value , must be collected using MPI and broadcast to each process, Store it in a temporary vector. It is a unit array. The calculation on the diagonal is , the remaining positions are :

[0067] (3)

[0068] (4) Shanghai Senburger Array H Front of column The element value is given by the formula (4) Calculate, then The value of the element is 0, It is a matrix-vector multiplication. Here we still use MPI to The partial element values ​​are collected and broadcast to each process:

[0069] (4)

[0070] (5) A set of standard orthogonal bases Basis vector The calculation is to multiply the matrix by the matrix first, and then multiply the matrix by the vector, adding a new Matrix multiplication, that is, formula (5) and (6). Broadcast to all processes, because Before OK The column is the unit matrix, so only Column to Multiply the columns and store them in a temporary matrix after each process. replace , is a unit vector whose jth element value is 1. The new Yes The jth column of

[0071] (5)

[0072] (6)

[0073] (6) The vector in formula (7) The calculation is also matrix multiplication, then multiplication by vector. According to the properties of the transformation matrix, , so we can omit the matrix multiplication and use Instead, calculate , which is convenient for using the data that has been stored by row , eliminating the need to divide by columns again:

[0074] (7)

[0075] Step 5: Parallel Givens Transformation

[0076] To make the residual The 2-norm of The smallest value is converted to The least squares problem is solved by QR decomposition of Givens transform. Givens transform has data dependence. Each process calculates the whole process in parallel. According to formulas (8), (9), and (10), the elements below the diagonal of the upper Hesenberg matrix H are changed to 0 each time. After m transformations, it becomes an upper triangular matrix, and According to formulas (11) and (12), we can transform the vector .

[0077] , (8)

[0078] (9)

[0079] (10)

[0080] (11)

[0081] (12)

[0082] in, , is the construction element of the Givens transformation.

[0083] Step 6: Solve the trigonometric equations in parallel to obtain new solutions and new residuals

[0084] Each process calculates the trigonometric equations in parallel, first calling the ztrsv function in the lapack library in parallel to perform trigonometric solutions and obtain the m-dimensional linear combination coefficients , each process obtains the solution according to the basis of the Krylov subspace ,Will Collect and summarize , and then calculate the new residual .

[0085] Step 7: Choose to continue iteration or terminate based on the judgment condition

[0086] 2 norm of residual and residual accuracy For comparison, if , the iteration terminates if , enter the next iteration , while updating and , until the residual requirement is met or the maximum number of iterations is reached. When each process calculates its own Inner product, each process is reduced and summed to get the total inner product, and the square root is obtained .

[0087] This embodiment tests the parallel iterative solution method for the electromagnetic finite element equations through experiments, specifically:

[0088] (1) Test environment: Sunward supercomputer (using Intel(R) Xeon(R) Gold 6258R processor)

[0089] (2) Test model: The number of electromagnetic linear equations is 1700, the subspace dimension is set to 1000, the initial solution is 0, and the accuracy is 10 -12 , the maximum number of iterations IterMax is 10000.

[0090] (3) Test results:

[0091] Figure 3 This is a comparison of the single-process running time and the time after different process optimization acceleration for solving the electromagnetic finite element equation group with a coefficient matrix dimension of 1700. It can be seen that the solution time before and after parallel optimization can be accelerated by up to 26.02 times, indicating that the Householder-GMRES parallel optimization for electromagnetic finite elements in this embodiment has exerted the powerful computing power of supercomputer multi-process and improved the solution speed of the sparse linear equation group of the electromagnetic field finite element method.

[0092] Example 2

[0093] In one embodiment of the present disclosure, a parallel iterative solution system for electromagnetic finite element equations is provided, comprising:

[0094] A building block configured to: build the electromagnetic finite element equations to be solved , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, and free space wave number; represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential;

[0095] The partitioning module is configured to: partition the electromagnetic finite element equations into multiple computational subtasks, and and the right-hand side vector Assigned to each process in the supercomputer, where Number the process;

[0096] The calculation module is configured as follows: using the initialized Householder-GMRES algorithm, each process performs subtask calculations in parallel to obtain the local optimal solution of each process ;

[0097] The aggregation module is configured to: Summarize and obtain the optimal solution of the electromagnetic finite element equations ;

[0098] The calculation of the subtask is an iterative calculation of Householder orthogonalization, Givens transformation and trigonometric equation solution in each process in turn until the condition for stopping the iteration is met and the local optimal solution is obtained. , in the iterative calculation process, the respective residuals , non-zero vector , unit vector , basis vector , a set of standard orthogonal basis , transformation matrix , Shanghai Senburger Array , local optimal solution A global inter-process communication protocol.

[0099] Example 3

[0100] In one embodiment of the present disclosure, a computer program product is provided, including a computer program, and when the computer program is executed by a processor, the parallel iterative solution method for electromagnetic finite element equations is implemented.

[0101] Example 4

[0102] In one embodiment of the present disclosure, a non-transitory computer-readable storage medium is provided, wherein the non-transitory computer-readable storage medium is used to store computer instructions. When the computer instructions are executed by a processor, the parallel iterative solution method for electromagnetic finite element equations is implemented.

[0103] Example 5

[0104] In one embodiment of the present disclosure, an electronic device is provided, including: a processor, a memory, and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory, so that the electronic device executes the parallel iterative solution method for electromagnetic finite element equations.

[0105] The present disclosure is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present disclosure. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of the processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0106] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0107] Although the above describes the specific implementation methods of the present disclosure in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present disclosure. Technical personnel in the relevant field should understand that on the basis of the technical solution of the present disclosure, various modifications or variations that can be made by those skilled in the art without creative work are still within the scope of protection of the present disclosure.

Claims

1. A parallel iterative solution method for electromagnetic finite element equations, characterized in that: include: Construct the electromagnetic finite element equations to be solved , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, and free space wave number; x represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential; Divide the electromagnetic finite element equations into multiple computational subtasks and block the submatrices of the subtasks and the right-hand side vector Assigned to each process in the supercomputer, where Number the process; Using the initialized Householder-GMRES algorithm, each process performs subtask calculations in parallel to obtain the local optimal solution of each process. ; Local optimal solution for each process Summarize and obtain the optimal solution of the electromagnetic finite element equations ; The calculation of the subtask is an iterative calculation of Householder orthogonalization, Givens transformation and trigonometric equation solution in each process in turn until the condition for stopping the iteration is met and the local optimal solution is obtained. , during the iterative calculation process, the local residual , non-zero vector , unit vector , basis vector , a set of standard orthogonal basis , transformation matrix , Shanghai Senburger Array , local optimal solution A global inter-process communication protocol.

2. A parallel iterative solution method for electromagnetic finite element equations according to claim 1, characterized in that: The division is to divide the equation groups into multiple computing subtasks evenly according to the number of equation groups, and each process undertakes its own computing subtask; if it cannot be completely evenly divided, the last process will undertake the remaining tasks.

3. A parallel iterative solution method for electromagnetic finite element equations according to claim 1, characterized in that: The Householder-GMRES algorithm has the following specific steps: By setting the initial parameters of Householder-GMRES; Through Householder orthogonalization, calculate the residual, unit vector, orthogonal basis matrix, transformation matrix and generate the Upper Heisenberg array; The Heisenberg array is transformed by Givens transformation Transform into an upper triangular matrix; Calculate new solutions and new residuals by solving trigonometric equations; Depending on the condition when the iteration stops, continue iterating or terminate and exit.

4. A parallel iterative solution method for electromagnetic finite element equations according to claim 1, characterized in that: The Householder orthogonalization is specifically: Based on the local solutions of the initial settings, the respective local residuals are calculated; Based on the local solution and local residual, Householder transformation is performed to calculate the unit vector, orthogonal basis matrix, transformation matrix and generate the upper Heisenberg array.

5. The parallel iterative solution method for electromagnetic finite element equations according to claim 1, characterized in that: The Givens transformation is specifically: The least squares problem of the subtask is solved by QR decomposition of Givens transformation. Each time, the elements below the diagonal of the upper Heisenberg array are changed to 0. After multiple transformations, it becomes an upper triangular matrix.

6. A parallel iterative solution method for electromagnetic finite element equations according to claim 1, characterized in that: The trigonometric equations are solved as follows: Each process first solves the least squares problem of the residual to obtain the linear combination coefficients, then obtains a new solution based on the basis of the Krylov subspace, and finally calculates the new residual.

7. A parallel iterative solution system for electromagnetic finite element equations, characterized in that: include: A building block configured to: build the electromagnetic finite element equations to be solved , where A is a sparse matrix of N rows and N columns, consisting of magnetic permeability, dielectric constant, and free space wave number; x represents the electric field intensity of the physical quantity to be solved, and b represents the physical quantity potential; The partitioning module is configured to: partition the electromagnetic finite element equations into multiple computational subtasks, and and the right-hand side vector Assigned to each process in the supercomputer, where Number the process; The calculation module is configured as follows: using the initialized Householder-GMRES algorithm, each process performs subtask calculations in parallel to obtain the local optimal solution of each process ; The aggregation module is configured to: Summarize and obtain the optimal solution of the electromagnetic finite element equations ; The calculation of the subtask is an iterative calculation of Householder orthogonalization, Givens transformation and trigonometric equation solution in each process in turn until the condition for stopping the iteration is met and the local optimal solution is obtained. , in the iterative calculation process, the respective residuals , non-zero vector , unit vector , basis vector , a set of standard orthogonal basis , transformation matrix , Shanghai Senburger Array , local optimal solution A global inter-process communication protocol.

8. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the parallel iterative solution method for electromagnetic finite element equations described in any one of claims 1 to 6 is implemented.

9. A non-transitory computer-readable storage medium, characterized in that: The non-transitory computer-readable storage medium is used to store computer instructions, and when the computer instructions are executed by the processor, a parallel iterative solution method for electromagnetic finite element equations as described in any one of claims 1 to 6 is implemented.

10. An electronic device, characterized in that: include: A processor, a memory and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory so that the electronic device executes a parallel iterative solution method for electromagnetic finite element equations as described in any one of claims 1 to 6.

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