Related papers: A Constraint energy minimizing generalized multisc…
The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order…
This paper aims to study the convergence of adaptive finite element method for control constrained elliptic optimal control problems under $L^2$-norm. We prove the contraction property and quasi-optimal complexity for the $L^2$-norm errors…
We propose an efficient and accurate parametric finite element method (PFEM) for solving sharp-interface continuum models for solid-state dewetting of thin films with anisotropic surface energies. The governing equations of the…
We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on…
In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second order elliptic equations in non-divergence form. The elliptic equation is casted into a…
We study the slightly compressible Darcy-Forchheimer equations modeling gas flow in porous media, particularly in applications related to combustion processes. The equations are discretized in time using the backward Euler method and in…
In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is…
Considering the increasing number of experimental results in the manufacturing process of quantum dots with different geometries, and the fact that most numerical methods that can be used to investigate quantum dots with non-trivial…
In this work, we extend the meshfree generalized multiscale exponential integration framework introduced in Nikiforov et al. (2025) to the simulation of three-dimensional advection--diffusion problems in heterogeneous and high-contrast…
We develop a sparse multiscale operator-adapted wavelet decomposition-based finite element method (FEM) on unstructured polygonal mesh hierarchies obtained via a coarsening procedure. Our approach decouples different resolution levels,…
We formulate and analyze a goal-oriented adaptive finite element method (GOAFEM) for a semilinear elliptic PDE and a linear goal functional. The strategy involves the finite element solution of a linearized dual problem, where the…
We present a detailed analysis of the bounds on the integration step in Discrete Element Method (DEM) for simulating collisions and shearing of granular assemblies. We show that, in the numerical scheme, the upper limit for the integration…
This work presents a practical finite element modeling strategy, the Crack Element Method (CEM), for simulating the dynamic crack propagation in two-dimensional structures. The method employs an element-splitting algorithm based on the…
In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which…
We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on…
In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous…
Simulating flow in a highly heterogeneous reservoir with multiscale characteristics could be considerably demanding. To tackle this problem, we propose a numerical scheme coupling the Generalized Multiscale Finite Element Method (GMsFEM)…
This work develops novel energy-stable parametric finite element methods (ES-PFEM) for the Willmore flow and curvature-dependent geometric gradient flows of surfaces in three dimensions. The key to achieving the energy stability lies in the…
Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…
In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schr\"{o}dinger equations with general multiscale potentials in the semiclassical regime. In the OC MsFEM the multiscale…