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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…
In this paper, we develop a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second-order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity numerical method adopts…
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on…
This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet…
This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial…
We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence,…
The solution of the elliptic partial differential equation has interface singularity at the points which are either the intersections of interfaces or the intersections of interfaces with the boundary of the domain. The singularities that…
Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a `parallel in real time'…
In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm…
A quasi-second order scheme is developed to obtain approximate solutions of the shallow water equationswith bathymetry. The scheme is based on a staggered finite volume scheme for the space discretization:the scalar unknowns are located in…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
In this paper, we consider the numerical approximation for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn-Hilliard equations, Navier-Stokes equations and…
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and…
Based on the work of Xu and Zhou [Math.Comput., 69(2000), pp.881-909], we establish new three-level and multilevel finite element discretizations by local defect-correction technique. Theoretical analysis and numerical experiments show that…
In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu…
Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full…
In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and…
We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress…
In this paper, we develop a numerical multiscale method to solve elliptic boundary value problems with heterogeneous diffusion coefficients and with singular source terms. When the diffusion coefficient is heterogeneous, this adds to the…
The oversampling multiscale finite element method (MsFEM) is one of the most popular methods for simulating composite materials and flows in porous media which may have many scales. But the method may be inapplicable or inefficient in some…