Related papers: A Stabilized bi-grid method for Allen Cahn equatio…
Finite Difference methods (FD) are one of the oldest and simplest methods for solving partial differential equations (PDE). Block Finite Difference methods (BFD) are FD methods in which the domain is divided into blocks, or cells,…
This article considers the extension of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when…
In this paper, we propose and analyze a time-stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for…
We present a stabilized, structure-preserving finite element framework for solving the Vlasov-Maxwell equations. The method uses a tensor product of continuous polynomial spaces for the spatial and velocity domains, respectively, to…
We introduce a new family of high order accurate semi-implicit schemes for the solution of non-linear hyperbolic partial differential equations on unstructured polygonal meshes. The time discretization is based on a splitting between…
We present a multiscale mixed finite element method for solving second order elliptic equations with general $L^{\infty}$-coefficients arising from flow in highly heterogeneous porous media. Our approach is based on a multiscale spectral…
Consistent splitting schemes are among the most accurate pressure segregation methods, incurring no splitting errors or spurious boundary conditions. Nevertheless, their theoretical properties are not yet fully understood, especially when…
We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in…
We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which…
A finite element scheme for an entirely fractional Allen-Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace…
The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions,…
This paper is concerned with mixed finite element method (FEM) for solving the two-dimensional, nonlinear fourth-order active fluid equations. By introducing an auxiliary variable $w=-\Delta u$, the original fourth problem is transformed…
We develop a conservative cut finite element method for an elliptic coupled bulk-interface problem. The method is based on a discontinuous Galerkin framework where stabilization is added in such a way that we retain conservation on macro…
This paper aims to develop a linearly implicit structure-preserving numerical scheme for the space fractional sine-Gordon equation, which is based on the newly developed invariant energy quadratization method. First, we reformulate the…
We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case in certain discrete…
This paper presents a first-order convex splitting hybridizable/embedded discontinuous Galerkin method for the phase field crystal equation written in mixed form. Since the sixth-order phase field crystal equation is rewritten as a…
We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite…
We present a novel structure-preserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the Lax-Friedrichs flux to a high-order staggered…