2 research outputs found
A Domain-Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients
We develop a domain-decomposition model reduction method for linear
steady-state convection-diffusion equations with random coefficients. Of
particular interest to this effort are the diffusion equations with random
diffusivities, and the convection-dominated transport equations with random
velocities. We investigate the equations with two types of random fields, i.e.,
colored noises and discrete white noises, both of which can lead to
high-dimensional parametric dependence. The motivation is to use domain
decomposition to exploit low-dimensional structures of local problems in the
sub-domains, such that the total number of expensive PDE solves can be greatly
reduced. Our objective is to develop an efficient model reduction method to
simultaneously handle high-dimensionality and irregular behaviors of the
stochastic PDEs under consideration. The advantages of our method lie in three
aspects: (i) online-offline decomposition, i.e., the online cost is independent
of the size of the triangle mesh; (ii) operator approximation for handling
non-affine and high-dimensional random fields; (iii) effective strategy to
capture irregular behaviors, e.g., sharp transitions of the PDE solution. Two
numerical examples will be provided to demonstrate the advantageous performance
of our method
A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems
A systematic numerical study on weak Galerkin (WG) finite element method for
second order linear parabolic problems is presented by allowing polynomial
approximations with various degrees for each local element. Convergence of both
semidiscrete and fully discrete WG solutions are established in
and norms for a general WG element , where
, and are arbitrary integers. The fully discrete
space-time discretization is based on a first order in time Euler scheme. Our
results are intended to extend the numerical analysis of WG methods for
elliptic problems [J. Sci. Comput., 74 (2018), 1369-1396] to parabolic
problems. Numerical experiments are reported to justify the robustness,
reliability and accuracy of the WG finite element method