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In this paper, we employ a space-time finite element method to discretize the parabolic initial-boundary value problem and extend its error analysis with refined estimates on unstructured space-time meshes. We establish higher-order…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
Linear stationary reaction-convection-diffusion equations with Dirichlet boundary conditions are approximated using a simple finite difference method corresponding to central differences and the addition of a high-order stabilization term…
The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations…
A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the…
We discuss the identification of a time-dependent potential in a time-fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem.…
By means of an original approach, called "method of the moving frame", we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path dependent…
In this work we are interested in effectively solving the quasi-static, linear Biot model for poromechanics. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well-known…
This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping…
We discretize the Lagrange multiplier formulation of the obstacle problem by mixed and stabilized finite element methods. A priori and a posteriori error estimates are derived and numerically verified.
We study a higher-order surface finite element (SFEM) penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated which are equivalent in the continuous setting. The impact of the…
In this paper, we develop the numerical theory of decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on…
One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations, is to render them from their hyperbolic character to elliptic form. This paper is devoted to the study of the stabilized linear…
We develop and analyze a stabilization term for cut finite element approximations of an elliptic second order partial differential equation on a surface embedded in $\mathbb{R}^d$. The new stabilization term combines properly scaled normal…
Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps…
In this paper, we carry out the error analysis for the structure-preserving discretization of the incompressible MHD system. This system, as a coupled system of Navier-Stokes equations and Maxwell's equations, is nonlinear. We use its…
In this paper, we formulate, analyse and implement the discrete formulation of the Brinkman problem with mixed boundary conditions, including slip boundary condition, using the Nitsche's technique for virtual element methods. The divergence…
We consider the systematic numerical approximation of Biot's quasistatic model for the consolidation of a poroelastic medium. Various discretization schemes have been analysed for this problem and inf-sup stable finite elements have been…
We present a space-time virtual element method for the discretization of the heat equation, which is defined on general prismatic meshes and variable degrees of accuracy. Strategies to handle efficiently the space-time mesh structure are…
Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques,…