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 |