Second-order backward difference stable generalized finite element method for parabolic interface problem

Abstract

The invention discloses a second-order backward difference stable generalized finite element method for a parabolic interface problem, and relates to the technical field of engineering application of stable generalized finite element methods, a stable generalized finite element method (SGFEM) is used for discretizing space variables, a second-order backward difference formula (BDF) is used for time discretization, and a stable generalized finite element method is used for discretizing space variables. A numerical value test verifies that the algorithm has the optimal convergence rate under an infinite norm and an L2 norm, a numerical value result shows that the algorithm shows unconditional convergence under a large time step, and a new solving strategy is provided for the parabolic equation interface problem.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of engineering application of stabilized generalized finite element method, and particularly to a second-order backward difference stabilized generalized finite element method for parabolic interface problems. BACKGROUND

[0002] In the field of engineering and material science, numerical simulation of fluid flow in fiber composites and porous media often involves two or more different materials or fluids, which is mathematically referred to as an interface problem. The low regularity of the exact solution of the model problem has a significant impact on the accuracy of the numerical method. Therefore, when solving interface problems using traditional finite element methods, it is difficult to find a suitable balance between obtaining satisfactory numerical accuracy and minimizing computational cost.

[0003] Recently, many experts in the field of scientific computing have devoted themselves to developing finite element methods for interface non-conforming meshes to solve parabolic interface problems, such as immersed finite element methods, extended finite element methods, etc. However, the traditional immersed finite element basis function may be discontinuous on the interface edge, which can have a negative impact on convergence. Professor Lin Tao et al. disclosed a partially penalized immersed finite element method in Partially penalized immersed finite element methods for parabolic interface problems and Optimal error bounds for partially penalized immersed finite element methods for parabolic interface problems to overcome these negative effects. However, due to the penalty parameter in the weak form, this partially penalized immersed finite element method is non-conforming.

[0004] In addition, in the numerical calculation of parabolic interface problems, the design of the time integration scheme and the design of the spatial discretization method are equally important. In the current research on parabolic interface problems, most of the efforts are devoted to the construction of spatial discretization methods for parabolic interface problems, while ignoring the time integration scheme. In the Analysis of backward Euler / extendedfinite element discretization of parabolic problems with moving interfaces, the authors mainly studied the backward Euler scheme and the Crank-Nicolson scheme for time discretization. These two numerical schemes only involve two time layers. The backward Euler scheme has only first-order time accuracy, while the Crank-Nicolson scheme has second-order time accuracy.

[0005] In order to solve the above problems, the present invention proposes a second-order backward difference stabilized generalized finite element method for the parabolic interface problem, which has no penalty terms and penalty parameters in the variational form and is applicable to large time steps. Summary of the Invention

[0006] In order to solve the above problems, the present invention aims to provide a second-order backward difference stabilized generalized finite element method for parabolic interface problems, comprising the following steps:

[0007] S10. Based on Green's formula and Dirichlet boundary conditions of the model problem, derive the variational weak form of the interface problem of parabolic equations;

[0008] S20. Spatial discretization format for constructing variational problems using the stable generalized finite element method;

[0009] S30. Use the second-order backward difference formula to construct a time-discrete format for variational problems;

[0010] S40. Design a program for the interface problem of parabolic equations based on the fully discrete variational format in a stable generalized finite element space.

[0011] Furthermore, in step S10, based on Green's formula and the boundary conditions of the model problem, a variational weak form of the interface problem of the parabolic equation is derived, specifically,

[0012] The parabolic interface problem with Dirichlet boundary conditions and initial value conditions is:

[0013]

[0014] Dirichlet boundary conditions:

[0015] Initial value condition: u(x, 0) = u0 (x), x∈Ω;

[0016] At the same time, the model problem satisfies two conditions at the interface Γ:

[0017]

[0018] Where Ω represents a piecewise smooth boundary Two-dimensional bounded convex region, smooth interface curve The bounded region Ω is divided into two disjoint regions Ω0 and Ω1, and the interface Γ does not intersect the boundary of the region Ω. The coefficient k(x) represents a piecewise constant function, that is, k(x) = k i >0, x∈Ω i , i=0,1, Representing boundaries The unit external normal vector of .

[0019] Under the Dirichlet boundary conditions and initial value conditions, according to the variational principle, given u 0 (x), find u(x)∈H 1 (Ω), and So that:

[0020]

[0021] The bilinear form is defined as:

[0022]

[0023] The linear form is defined as:

[0024] L(w)=f Ω fwdx.

[0025] Furthermore, the step S20, constructing a spatial discretization format of the variational problem using the stable generalized finite element method, includes:

[0026] Given a bounded region Ω and a mesh parameter h, the standard finite element meshing on the region Ω is where K i represents the i-th quadrilateral unit, and Λ is the total number of units. N i (x),i∈I h Represents the grid node x i The standard finite element shape function at , where I h Represents the set of all node numbers of the mesh.

[0027] Generalized finite element space S GFEM is the standard finite element space S FEM and the non-polynomial enrichment space S ENR The direct sum of the enriched space SENR are composed of the product of traditional finite element basis functions and non-polynomial enrichment functions. The approximation functions in the generalized finite element space S GFEM are expressed as:

[0028]

[0029] wherein, denotes the enrichment node index set, ψ denotes the enrichment function, simulates the non-smooth characteristics of the interface problem, c i denotes the standard finite element free degree of the node x i , a j denotes the enrichment free degree of the node x j .

[0030] The selection of the enrichment function depends on the non-smooth characteristics of the model problem. The generalized finite element method may produce a bad condition number and may not achieve the optimal convergence order. In order to improve the matrix condition number and eliminate the error influence caused by the mixed element.

[0031] Therefore, the step S20, the construction of the variational problem space discrete format of the stabilized generalized finite element method, further comprises: adopting the revised enrichment function to stabilize the generalized finite element method,

[0032] The approximate solution in the stabilized generalized finite element space S SGFEM is expressed as:

[0033]

[0034] wherein, the index set The revised enrichment function ψ = ψ - Π h ψ, Π h ψ is the interpolation of the function ψ in the standard finite element space;

[0035] In order to simulate weak discontinuity (the function is continuous, but its derivative is discontinuous), the distance function or the level set function is selected as the enrichment function, and the distance function is selected as the enrichment function:

[0036]

[0037] Or the absolute value of the level set function is selected as the enrichment function:

[0038]

[0039] wherein, is the level set function;

[0040] In the present application, the one-sided distance function is selected to construct the revised enrichment function:

[0041]

[0042] The stable generalized finite element method (SGM) offers excellent numerical performance for solving interface problems, demonstrating good convergence, stability, and robustness. The algorithm achieves optimal convergence in the energy norm; the growth order of the scalarized condition number is consistent with that of the standard finite element method; and the stiffness and mass matrix condition numbers remain essentially stable when the interface is close to the mesh boundary.

[0043] Furthermore, the step S30, constructing a time discrete format of the variational problem using a second-order backward difference formula, includes:

[0044] For a given time interval [0, T] and a positive integer N, mesh it {t n :t n =nτ,0≤n≤N}, where τ is the time step, and the function sequence {u n :u n =u(x,t n ),0≤n≤N}, then the second-order backward difference format (BDF2) is:

[0045]

[0046] The time discretization format has second-order time accuracy. The full discretization format for the parabolic interface problem is: turn up So that:

[0047]

[0048] in,

[0049]

[0050] To stabilize the generalized finite element space S SGFEM The two subspaces of , the discrete linear form is:

[0051] L n (w h )=f Ω f(xt n )w h dx;

[0052] Without any boundary conditions, the discrete matrix form of the variational form is derived:

[0053]

[0054] in,

[0055]

[0056] K is the stiffness matrix, M is the mass matrix, L nis the load vector.

[0057] Furthermore, the step S30 of constructing the time discrete format of the variational problem using the second-order backward difference formula further includes: applying a non-homogeneous Dirichlet boundary condition to the discrete matrix form. Because the interface and the boundary do not intersect, the boundary condition is directly applied to the standard finite element part of the stable generalized finite element solution. The boundary condition is:

[0058]

[0059] make:

[0060]

[0061] Represents the node index set on the region boundary, assuming U n A component c k is known, that is Note the system matrix Right vector Perform the following steps:

[0062] S31, C k，j =0,

[0063] S32, C k，k =1;

[0064] S33,

[0065] Then U n The component c k The non-homogeneous Dirichlet boundary conditions have been successfully imposed on all boundary node index sets Repeating the above steps, the non-homogeneous Dirichlet boundary conditions have been fully integrated into the discrete matrix form of the variational form.

[0066] Furthermore, in step S40, a program is designed for the interface problem of the parabolic equation based on the fully discrete variational format in the stable generalized finite element space. Due to the non-smooth characteristics of the basis functions in the enriched space, according to the situation where the interface passes through the unit, a straight line is used instead of a curve in the interface unit to segment the interface unit, so as to perform Gaussian numerical integration on the stiffness matrix, mass matrix, right-hand side terms and boundary condition terms in the discrete variational format, thereby deriving the relevant matrices and right-hand side terms, and applying non-homogeneous boundary conditions to solve the problem.

[0067] The specific steps include:

[0068] S41. Given a model problem and interface, discretize the computational domain Ω and the time interval [0, T];

[0069] S42. Cut the interface unit into 4-6 triangular sub-regions, perform Gaussian numerical integration on the stiffness matrix K and mass matrix M, and synthesize the system matrix

[0070] S43, according to the initial value condition u 0 (x) Calculate its stability in the generalized finite element space S SGFEM L 2 projection At the same time calculate

[0071] S44, according to and Perform time layer iteration and calculate the load vector L according to the numerical integration scheme n , then synthesize the right end vector Finally, Dirichlet boundary conditions are applied to the discrete matrix form to calculate 2≤n≤N;

[0072] S45, calculation Relative error norm for 2≤n≤N, including infinity norm and L 2 norm.

[0073] Furthermore, the Gaussian numerical integration in step S42 has a specific integration scheme as follows: for non-interface units, a standard 4×4 Gaussian integration rule is used for numerical integration; for triangular sub-regions of interface units, a 6×6 Gaussian integration rule is used for numerical integration.

[0074] Furthermore, the infinite norm in step S45 is calculated by taking 10×10 sample points in each unit K and calculating the maximum value of all sample points in the area.

[0075] The beneficial effects of the present invention are as follows: the present invention constructs a stable generalized finite element method for discrete spatial variables, and the classic second-order backward difference formula is used for time discretization. The stable generalized finite element method adopts a unilateral enrichment function based on a simple fixed grid, and there is no penalty term and penalty parameter in the variational form. Numerical experiments verify that the stable generalized finite element method of the second-order backward difference formula can achieve infinite norm and L for solving parabolic interface problems. 2 The optimal convergence order under the norm is obtained, and the algorithm is applicable to large time steps, that is, the algorithm is unconditionally stable. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 Schematic diagram of Gaussian integration points of non-interface unit and interface unit;

[0077] Figure 2Schematic diagram of uniformly distributed sampling points on unit K. DETAILED DESCRIPTION

[0078] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. The exemplary embodiments and descriptions of the present invention are used to explain the present invention but are not intended to limit the present invention.

[0079] The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems includes the following steps:

[0080] S10. Based on Green's formula and the Dirichlet boundary condition of the model problem, derive the variational weak form of the interface problem of the parabolic equation, specifically,

[0081] The parabolic interface problem with Dirichlet boundary conditions and initial value conditions is:

[0082]

[0083] Dirichlet boundary conditions:

[0084] Initial value condition: u(x, 0) = u 0 (x), x∈Ω;

[0085] At the same time, the model problem satisfies two conditions at the interface Γ:

[0086]

[0087] Where Ω represents a piecewise smooth boundary Two-dimensional bounded convex region, smooth interface curve The bounded region Ω is divided into two disjoint regions Ω0 and Ω1, and the interface Γ does not intersect the boundary of the region Ω. The coefficient k(x) represents a piecewise constant function, that is, k(x) = k i >0, x∈Ω i , i=0,1, Representing boundaries The unit external normal vector of .

[0088] Under the Dirichlet boundary conditions and initial value conditions, according to the variational principle, given u 0 (x), find u(x)∈H 1 (Ω), and So that:

[0089]

[0090] The bilinear form is defined as:

[0091]

[0092] The linear form is defined as:

[0093] L(w)=f Ω fwdx.

[0094] S20. The spatial discretization format for constructing variational problems using the stable generalized finite element method includes:

[0095] Given a bounded region Ω and a mesh parameter h, the standard finite element meshing on the region Ω is where K i represents the i-th quadrilateral unit, and Λ is the total number of units. N i (x),i∈I h Represents the grid node x i The standard finite element shape function at , where I h Represents the set of all node numbers of the mesh.

[0096] Generalized finite element space S GFEM is the standard finite element space S FEM and the non-polynomial enrichment space S ENR The direct sum of the enriched space S ENR It is composed of the product of traditional finite element basis functions and non-polynomial enrichment functions. GFEM The approximation function in is expressed as:

[0097]

[0098] in, represents the enriched node index set, ψ represents the enrichment function, simulating the non-smooth characteristics of the interface problem, c i Represents node x i The standard finite element degree of freedom, a j Represents node x j degrees of freedom for enrichment.

[0099] The selection of the enrichment function depends on the non-smooth characteristics of the model problem. The generalized finite element method may produce bad condition numbers and may not reach the optimal convergence order. In order to improve the matrix condition number and eliminate the error caused by the mixed element, the enrichment function is used.

[0100] Therefore, the revised enrichment function stabilized generalized finite element method is adopted.

[0101] Stable generalized finite element space S SGFEM The approximate solution in is expressed as:

[0102]

[0103] Among them, the indicator set Revised enrichment function ψ = ψ - Π hψ,Π h ψ is the interpolation of the function ψ in the standard finite element space;

[0104] In order to simulate weak discontinuities (the function is continuous, but its derivative is discontinuous), we select the distance function or the level set function as the enrichment function. We select the distance function as the enrichment function:

[0105]

[0106] Or choose the absolute value of the level set function as the enrichment function:

[0107]

[0108] in, is the level set function;

[0109] In the present invention, a unilateral distance function is selected to construct a revised enrichment function:

[0110]

[0111] The stable generalized finite element method (SGM) offers excellent numerical performance for solving interface problems, demonstrating good convergence, stability, and robustness. The algorithm achieves optimal convergence in the energy norm; the growth order of the scalarized condition number is consistent with that of the standard finite element method; and the stiffness and mass matrix condition numbers remain essentially stable when the interface is close to the mesh boundary.

[0112] S30. Use the second-order backward difference formula to construct a time-discrete format for the variational problem, including:

[0113] For a given time interval [0, T] and a positive integer N, mesh it {t n :t n =nτ,0≤n≤N}, where τ is the time step, and the function sequence {u n :u n =u(x,t n ),0≤n≤N}, then the second-order backward difference format (BDF2) is:

[0114]

[0115] The time discretization format has second-order time accuracy. The full discretization format for the parabolic interface problem is: turn up So that:

[0116]

[0117] in,

[0118]

[0119] To stabilize the two subspaces of the generalized finite element space S SGFEM , the discrete linear form is:

[0120] L n (w h )=f Ω f(x,t n )w h dx;

[0121] Without any boundary conditions, the discrete matrix form of the variational form is derived:

[0122]

[0123] where,

[0124]

[0125] K is the stiffness matrix, M is the mass matrix, L n is the load vector.

[0126] Non-homogeneous Dirichlet boundary conditions are imposed on the discrete matrix form. Since the interface does not intersect the boundary, the boundary conditions are directly imposed on the standard finite element part of the stabilized generalized finite element solution, which are:

[0127]

[0128] Let:

[0129]

[0130] denote the set of node indices on the boundary of the domain, and assume that one component c n of U k is known, i.e. denote the system matrix and the right-hand side vector Perform the following steps:

[0131] S31, C k，j = 0,

[0132] S32, C k，k = 1;

[0133] S33,

[0134] then the component c n of U k has successfully imposed the non-homogeneous Dirichlet boundary conditions on all boundary node index sets Repeating the above steps, the non-homogeneous Dirichlet boundary conditions have been fully integrated into the discrete matrix form of the variational form.

[0135] S40. Design a program for the interface problem of parabolic equations based on the fully discrete variational format in a stable generalized finite element space.

[0136] Due to the non-smooth characteristics of the basis functions in the enriched space, straight lines are used instead of curves in the interface elements to segment the interface elements according to the situation where the interface passes through the elements. Gaussian numerical integration is performed on the stiffness matrix, mass matrix, right-hand side terms and boundary condition terms in the discrete variational format, thereby deriving the relevant matrices and right-hand side terms, and applying non-homogeneous boundary conditions for solution.

[0137] The specific steps include:

[0138] S41. Given a model problem and interface, discretize the computational domain Ω and the time interval [0, T];

[0139] S42. Cut the interface unit into 4-6 triangular sub-regions, perform Gaussian numerical integration on the stiffness matrix K and mass matrix M, and synthesize the system matrix The specific integration scheme is: for non-interface units, the standard 4×4 Gaussian integration rule is used for numerical integration; for the triangular sub-region of the interface unit, the 6×6 Gaussian integration rule is used for numerical integration. Figure 1 , Gaussian integration points of non-interface elements and interface elements, (a): non-interface elements; (b) and (c): interface elements, where the interface elements only list two interface cutting situations.

[0140] S43, according to the initial value condition u 0 (x) Calculate its stability in the generalized finite element space S SGFEM L 2 projection At the same time calculate

[0141] S44, according to and Perform time layer iteration and calculate the load vector L according to the numerical integration scheme n , then synthesize the right end vector Finally, Dirichlet boundary conditions are applied to the discrete matrix form to calculate 2≤n≤N;

[0142] S45, calculation Relative error norm for 2≤n≤N, including infinity norm and L 2The calculation method of the infinite norm is: take 10×10 sample points on each unit K, calculate the maximum value of all sample points in the area, and the uniformly distributed sampling points on the unit K are as follows: Figure 2 shown.

[0143] In order to evaluate the performance of the second-order backward difference formula-stable generalized finite element method (BDF2-SGFEM) and prove its beneficial effects, a number of numerical tests on diffusion coefficients and interface shapes were conducted, and the excerpts are as follows. The basic settings in the experiment are: Assume that the calculation area Ω = [a, b] 2 , interface Γ={x∈Ω:γ(x)=0}, subregion Ω0={x∈Ω:γ(x)<0}, subregion Ω1={x∈Ω:γ(x)>0}. The region W is divided into a J×J rectangular grid with grid parameters The time interval [0, T] is evenly divided into N subintervals, where the time step is T = 1. Two different sets of contrast coefficients are tested: (a) k0 = 10, k1 = 1; (b) k0 = 100, k1 = 1. The relative error norm is calculated to evaluate the optimal convergence and unconditional stability of the proposed numerical algorithm.

[0144] Test Example 1

[0145] Hypothetical interface Where x0=1 / 2, y0=1 / 2, The calculation area is Ω=[0,1] 2 , the analytical solution to the parabolic interface problem is:

[0146]

[0147] in Functions f and g can be determined by the above analytical solution.

[0148] In Experiment 1, BDF2-SGFEM and BDF2-FEM were used to calculate the infinity norm and L 2 The relative error of the norm is shown in Tables 1 to 4.

[0149] Table 1 Infinity norm and L for circular interface problem solved by BDF2-SGFEM 2 Norm error, τ = 4h

[0150]

[0151] Table 2 Infinity norm and L for circular interface problem solved by BDF2-SGFEM 2 Norm error, τ = 8h

[0152]

[0153] Tables 1 and 2 give the relative error and convergence order of BDF2-SGFEM when τ = 4h, 8h and different diffusion coefficients. It can be seen from the table that BDF2-SGFEM has good convergence in the infinite norm and L 2 The optimal second-order convergence can be achieved under the infinite norm. From Tables 3 and 4, we can see that BDF2-FEM has good performance in the infinite norm and L 2 Norm can only achieve suboptimal first-order convergence. As can be seen from Tables 1 and 2, BDF2-SGFEM can also converge to the exact solution with the optimal convergence order under large time steps, which shows that BDF2-SGFEM is unconditionally stable.

[0154] Table 3 Infinity norm and L for circular interface problem solved by BDF2-FEM 2 Norm error, τ = 4h

[0155]

[0156]

[0157] Table 4 Infinity norm and L for circular interface problem solved by BDF2-FEM 2 Norm error, τ = 8h

[0158]

[0159] Test Example 2

[0160] Assume that the interface of the parabolic interface problem is an elliptical interface Where α=π / 6.28, β=π / 9.42, and the calculation area is Ω=[-1,1] 2 , the analytical solution to the parabolic interface problem is:

[0161]

[0162] In Experiment 2, BDF2-SGFEM and BDF2-FEM are used to calculate the infinity norm and L 2 The relative error of the norm is shown in Tables 5 to 8.

[0163] Tables 5 and 6 give the relative errors and convergence orders for τ = 4h, 8h and different diffusion coefficients. As can be seen from the table, BDF2-SGFEM has good performance in terms of infinite norm and L 2 The optimal second-order convergence can be achieved under the infinite norm. From Tables 7 and 8, we can see that BDF2-FEM has good performance in the infinite norm and L 2 Norm can only achieve suboptimal first-order convergence. As can be seen from Tables 5 and 6, BDF2-SGFEM can also converge to the exact solution with the optimal convergence order under large time steps, which shows that BDF2-SGFEM is unconditionally stable.

[0164] Table 5 Infinity norm and L for BDF2-SGFEM solving elliptical interface problems 2 Norm error, τ = 4h

[0165]

[0166] Table 6 Infinity norm and L for BDF2-SGFEM solving elliptical interface problems 2 Norm error, τ = 8h

[0167]

[0168] Table 7 Infinity norm and L for BDF2-FEM solution of elliptical interface problem 2 Norm error, τ = 4h

[0169]

[0170] Table 8 Infinity norm and L for BDF2-FEM solution of elliptical interface problem 2 Norm error, τ = 8h

[0171]

[0172]

[0173] Conclusion: The second-order backward difference stable generalized finite element method for parabolic interface problems proposed in this paper is used to discretize spatial variables, while the classical second-order backward difference formula is used for time discretization. The stable generalized finite element method uses a unilateral enrichment function based on a simple fixed grid, and there is no penalty term or penalty parameter in the variational form. Numerical experiments verify that the second-order backward difference formula stable generalized finite element method can achieve infinite norm and L for solving parabolic interface problems. 2 The optimal convergence order under the L norm is achieved, and the algorithm is suitable for large time steps, that is, the algorithm is unconditionally stable. However, when the standard finite element method is used for spatial discretization, BDF2-FEM cannot achieve the infinite norm and L 2 Optimal convergence under the norm.

[0174] Although the present invention has been described in detail above using general explanations, specific embodiments, and experiments, it will be apparent to those skilled in the art that modifications and improvements may be made based on the present invention. Therefore, such modifications and improvements, which do not depart from the spirit of the present invention, are intended to be within the scope of protection claimed herein.

Claims

1. A second-order backward-difference stabilized generalized finite element method for parabolic interface problems, characterized by: The following steps are involved: S10. Based on Green's formula and Dirichlet boundary conditions of the model problem, derive the variational weak form of the interface problem of parabolic equations; S20. Spatial discretization format for constructing variational problems using the stable generalized finite element method; S30. Use the second-order backward difference formula to construct a time-discrete format for variational problems; S40. Design a program for the interface problem of parabolic equations based on the fully discrete variational format in a stable generalized finite element space.

2. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 1, characterized in that: The step S10, based on Green's formula and the boundary conditions of the model problem, derives the variational weak form of the parabolic equation interface problem, specifically, The parabolic interface problem with Dirichlet boundary conditions and initial value conditions is: Dirichlet boundary conditions: Initial value condition: u(x, 0) = u 0 (x), x∈Ω; At the same time, the model problem satisfies two conditions at the interface Γ: Where Ω represents a piecewise smooth boundary Two-dimensional bounded convex region, smooth interface curve The bounded region Ω is divided into two disjoint regions Ω0 and Ω1, and the interface Γ does not intersect the boundary of the region Ω. The coefficient k(x) represents a piecewise constant function, that is, k(x) = k i >0, x∈Ω i , i=0,1, Representing boundaries The unit external normal vector of ; Under the Dirichlet boundary conditions and initial value conditions, according to the variational principle, given u 0 (x), find u(x)∈H 1 (Ω), and So that: The bilinear form is defined as: The linear form is defined as: L(w)=∫ Ω fwdx。 3. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 1, characterized in that: The step S20, constructing a spatial discretization format for the variational problem using the stable generalized finite element method, includes: Given a bounded region Ω and a mesh parameter h, the standard finite element meshing on the region Ω is where K i represents the i-th quadrilateral unit, Λ is the total number of units, N i (x), i∈I h Represents the grid node x i The standard finite element shape function at , where I h Represents the set of all node numbers of the grid; Generalized finite element space S GFEM The approximation function in is expressed as: in, represents the enriched node index set, ψ represents the enrichment function, simulating the non-smooth characteristics of the interface problem, c i Represents node x i The standard finite element degree of freedom, a j Represents node x j degrees of freedom for enrichment.

4. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 3, characterized in that: The step S20, constructing a spatial discretization format of the variational problem using the stabilized generalized finite element method, further includes: using a revised enrichment function to stabilize the generalized finite element method, specifically: Stable generalized finite element space S SGFEM The approximate solution in is expressed as: Among them, the indicator set Revised enrichment function ∏ h ψ is the interpolation of the function ψ in the standard finite element space; Select the one-sided distance function to construct the revised enrichment function:

5. The second-order backward-difference stabilized generalized finite element method for parabolic interface problems according to claim 1, characterized in that: The step S30, constructing a time discrete format of the variational problem using a second-order backward difference formula, includes: For a given time interval [0, T] and a positive integer N, mesh it {t n :t n =nτ, 0≤n≤N}, where τ is the time step, and the function sequence {u n :u n =u(x, t n ), 0≤n≤N}, then the second-order backward difference format is: The full discretization format of the parabolic interface problem is: given turn up So that: in, To stabilize the generalized finite element space S SGFEM The two subspaces of , the discrete linear form is: L n (w h )=∫ Ω f(x,t n )w h dx; Without any boundary conditions, the discrete matrix form of the variational form is derived: in, K is the stiffness matrix, M is the mass matrix, L n is the load vector.

6. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 5, characterized in that: The step S30, constructing the time discrete format of the variational problem using the second-order backward difference formula, further includes: applying a non-homogeneous Dirichlet boundary condition to the discrete matrix form, where the interface and the boundary do not intersect, and the boundary condition is directly applied to the standard finite element part of the stable generalized finite element solution, and the boundary condition is: make: Represents the node index set on the region boundary, assuming U n A component c k is known, that is Note the system matrix Right vector Perform the following steps: S31、 S32、C k，k =1; S33, Then U n The component c k The non-homogeneous Dirichlet boundary conditions have been successfully imposed on all boundary node index sets Repeating the above steps, the non-homogeneous Dirichlet boundary conditions have been fully integrated into the discrete matrix form of the variational form.

7. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 1, characterized in that: The step S40, programming the interface problem of the parabolic equation according to the fully discrete variational format in the stable generalized finite element space, comprises the following steps: S41. Given a model problem and interface, discretize the computational domain Ω and the time interval [0, T]; S42. Cut the interface unit into 4-6 triangular sub-regions, perform Gaussian numerical integration on the stiffness matrix K and mass matrix M, and synthesize the system matrix S43, according to the initial value condition u 0 (x) Calculate its stability in the generalized finite element space S SGFEM L 2 projection At the same time calculate S44, according to and Perform time layer iteration and calculate the load vector L according to the numerical integration scheme n , then synthesize the right end vector Finally, Dirichlet boundary conditions are applied to the discrete matrix form to calculate S45, calculation The relative error norm of L, including the infinity norm and L 2 norm.

8. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 7, characterized in that: The Gaussian numerical integration in step S42 has the following specific integration schemes: for non-interface units, a standard 4×4 Gaussian integration rule is used for numerical integration; for triangular sub-regions of interface units, a 6×6 Gaussian integration rule is used for numerical integration.

9. The second-order backward-difference-stabilized generalized finite element method for parabolic interface problems according to claim 7, characterized in that: The calculation method of the infinite norm in step S45 is: take 10×10 sample points in each unit K and calculate the maximum value of all sample points in the area.

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