This work investigates the lowest-order weak Galerkin nite element (WGFE) method for solving reaction-diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well-posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h^0.5) and O(h) in H1 and L2 norms, respectively.

We propose a unified framework for the construction of the schemes and analysis of the L2 error estimate for arbitrary k order finite volume element method (FVEM) on triangular meshes. “orthogonal conditions” are proposed to construct the dual meshes, which guarantee the optimal convergence rate with L2 norm of the corresponding FVEM schemes. Moreover, with the orthogonal conditions, we establish the superconvergence theory for quadratic FVEM on triangular meshes.

In this article, a weak Galerkin finite element method for Laplace equation using harmonic polynomial space is proposed and analyzed. The idea of using \(P_k\)-harmonic polynomial space instead of the full polynomial space \(P_{k}\) is to use a much smaller number of basis functions to achieve same accuracy when \(k >= 2\). The optimal rate of convergence is derived in both \(H^1\) and \(L^2\) norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using \(P_{2}\)-harmonic polynomial space and using the standard \(P_2\) polynomial space are presented.

In this talk, we present an efficient and robust numerical solver for anisotropic subdiffusion problems, which have not been fully investigated in the literature. The Chebyshev spectral collocation method is utilized for discretization of the spatial Laplacian, whereas a linear interpolation is used for discretizing the fractional order Caputo temporal derivative. This solver is stable and catches the main features of subdiffusion. Numerical experiments are presented to demonstrate the accuracy and efficiency of this new solver.

This presentation presents a fully-kinetic numerical investigation of plasma charging at lunar carters using the most recently developed Parallel Immersed-Finite-Element Particle-in-Cell (PIFE-PIC) code. This model explicitly includes the lunar regolith layer and the bedrock in the simulation domain, taking into account of the regolith layer thickness and permittivity, and is capable of resolving a non trivial surface terrain or spacecraft configuration. 3-D domain decomposition is used for both particle-push and field-solve steps of the PIC loop to distribute the computation among multiple processors. Simulations are presented to study surface charging and lunar lander charging near the lunar craters.

We propose and analyze an efficient ensemble algorithm with artificial compressibility for fast decoupled computation of multiple realizations of the stochastic Stokes-Darcy model with random hydraulic conductivity (including the one in the interface conditions), source terms, and initial conditions. The solutions are found by solving three smaller decoupled subproblems with two common time-independent coefficient matrices for all realizations, which significantly improves the efficiency for both assembling and solving the matrix systems. The fully coupled Stokes-Darcy system can be first decoupled into two smaller sub-physics problems by the idea of the partitioned time stepping, which reduces the size of the linear systems and allows parallel computing for each sub-physics problem. The artificial compressibility further decouples the velocity and pressure which further reduces storage requirements and improves computational efficiency. We prove the long time stability and the convergence for this new ensemble method.

When using proper orthogonal decomposition (POD) as a model order reduction technique it is of particular interest to know how these approximations behave. We present refinements and extensions of our earlier results concerning POD projections and approximation errors. We focus on new, exact error formulas for POD approximations with a generalized framework.

In this presentation, different finite element methods with discontinuous approximations will be discussed including IPDG, HDG and specially WG finite element methods as well as the relations between them. In addition, new stabilizer free discontinuous finite element methods will be introduced.