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ABSTRACT
This paper presents a fast and stabilized meshless method that combines variational multi-scale element free Galerkin (VMEFG) method and proper orthogonal decomposition (POD) method, namely VMEFG&POD, to solve convection-diffusion problems. Variational multi-scale method is applied to overcome the numerical oscillation for the convection-dominated problems and the POD method is used to improve the computational efficiency of the VMEFG method. This method is validated by considering the simulation of two-dimensional convection-diffusion problems with a small diffusion coefficient. It is demonstrated that the proposed method can largely improve the computational efficiency without a significant loss in accuracy.
Nomenclature
| a | = | known convection velocity |
| b1 | = | bubble function for the trial solution |
| b2 | = | bubble function for the weighting function |
| d | = | the degree of freedom of the model |
| e(l) | = | the error of energy |
| f | = | known source term |
| F | = | the right-hand side vector |
| = | reduced load vector of F | |
| m | = | the number of snapshots |
| M | = | mass matrix |
| = | reduced system matrix of M | |
| k | = | diffusion coefficient |
| K | = | a matrix related to the space |
| = | reduced system matrix of K | |
| l | = | the number of singular values |
| L2 | = | error norm |
| S | = | a rectangular m × d matrix |
| u | = | unknown field variable |
| u(t) | = | an unknown vector |
| usnap | = | a rectangular m × d matrix |
| = | reduced new unknown vector | |
| = | trial function in coarse scale | |
| u′(t, x) | = | trial function in fine scale |
| V | = | the space of weighting functions |
| V1 | = | orthogonal matrix |
| V2 | = | orthogonal matrix |
| w | = | weighting function |
| W | = | coarse-scale problem |
| W′ | = | fine-scale problem |
| = | weighting function in coarse scale | |
| w′(t, x) | = | weighting function in fine scale |
| Ω | = | bounded domain |
| ∂Ω | = | the boundary of the bounded domain |
| ΩK | = | background integral cell |
| = | the residual of equation | |
| τ | = | the stability parameter of VMEFG method |
| σ | = | reaction coefficient |
| λi | = | singular values of matrixusnap |
| Φ | = | optimal orthonormal basis |
| ψ(t, x) | = | the prescribed boundary condition |
| Δt | = | time-step |
Nomenclature
| a | = | known convection velocity |
| b1 | = | bubble function for the trial solution |
| b2 | = | bubble function for the weighting function |
| d | = | the degree of freedom of the model |
| e(l) | = | the error of energy |
| f | = | known source term |
| F | = | the right-hand side vector |
| = | reduced load vector of F | |
| m | = | the number of snapshots |
| M | = | mass matrix |
| = | reduced system matrix of M | |
| k | = | diffusion coefficient |
| K | = | a matrix related to the space |
| = | reduced system matrix of K | |
| l | = | the number of singular values |
| L2 | = | error norm |
| S | = | a rectangular m × d matrix |
| u | = | unknown field variable |
| u(t) | = | an unknown vector |
| usnap | = | a rectangular m × d matrix |
| = | reduced new unknown vector | |
| = | trial function in coarse scale | |
| u′(t, x) | = | trial function in fine scale |
| V | = | the space of weighting functions |
| V1 | = | orthogonal matrix |
| V2 | = | orthogonal matrix |
| w | = | weighting function |
| W | = | coarse-scale problem |
| W′ | = | fine-scale problem |
| = | weighting function in coarse scale | |
| w′(t, x) | = | weighting function in fine scale |
| Ω | = | bounded domain |
| ∂Ω | = | the boundary of the bounded domain |
| ΩK | = | background integral cell |
| = | the residual of equation | |
| τ | = | the stability parameter of VMEFG method |
| σ | = | reaction coefficient |
| λi | = | singular values of matrixusnap |
| Φ | = | optimal orthonormal basis |
| ψ(t, x) | = | the prescribed boundary condition |
| Δt | = | time-step |