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We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion…
We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of…
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…
This is a study of certain finite element methods designed for convection-dominated, time-dependent partial differential equations. Specifically, we analyze high order space-time tensor product finite element discretizations, used in a…
A Streamline Upwind Petrov-Galerkin (SUPG) finite element method for transient convection-diffusion-reaction equation in time-dependent domains is proposed. In particular, a convection dominated transient scalar problem is considered. The…
This paper explores a fully discrete approximation for a nonlinear hyperbolic PDE-constrained optimization problem (P) with applications in acoustic full waveform inversion. The optimization problem is primarily complicated by the…
In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet number. In this situation, computational instabilities occur, both for steady and…
This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure…
Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…
This paper concerns with finite element approximations of a quasi-static poroelasticity model in displacement-pressure formulation which describes the dynamics of poro-elastic materials under an applied mechanical force on the boundary. To…
The derivation of second-order ordinary differential equations (ODEs) as continuous-time limits of optimization algorithms has been shown to be an effective tool for the analysis of these algorithms. Additionally, discretizing…
A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive…
We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in an Eulerian-Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical formulation adapts 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 construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As origin, we consider a reformulation of the system in order to decouple the discretization…
In (J. Comput. Phys., 417, 109577, 2020) we introduced a space-time embedded-hybridizable discontinuous Galerkin method for the solution of the incompressible Navier-Stokes equations on time-dependent domains of which the motion of the…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the…
We consider a time-dependent coupled Navier--Stokes/generalized poroelastic flow problem and propose a unified and monolithic finite element discretization based on implicit time stepping. To handle the fluid-structure interface we employ a…
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…