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This article provides techniques of raising the regularity of fractional order equations and resolves fundamental questions on the one-dimensional homogeneous boundary-value problem of skewed (double-sided) fractional diffusion advection…
In this article we develop the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the…
We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The…
This article presents a new finite element method for convection-diffusion equations by enhancing the continuous finite element space with a flux space for flux approximations that preserve the important mass conservation locally on each…
We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of…
We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show…
A new method recovers the phase difference of interfering wavefronts from a pattern of interference fringes, avoiding the phase discontinuity problem. The method relies on the numerical solution of one-dimensional first-order ordinary…
Within this chapter, we discuss control in the coefficients of an obstacle problem. Utilizing tools from H-convergence, we show existence of optimal solutions. First order necessary optimality conditions are obtained after deriving…
An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality…
We develop a spectral low-mode reduced solver for second-order elliptic boundary value problems with spatially varying diffusion coefficients. The approach projects standard finite difference or finite element discretization onto a global…
An approximate method is proposed for the recovery of a compactly supported spherically-symmetric potential from the set of fixed-energy phase-shifts known for all angular momenta. The method reduces the inverse scattering problem to a…
We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show…
We study the numerical approximation of advection-diffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method. The latter method is a now classical, finite…
We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting non-linear…
In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks. The numerical scheme is based on the standard output least-squares formulation where the…
Spectral asymptotics of linear periodic elliptic operators with indefinite (sign-changing) density function is investigated in perforated domains with the two-scale convergence method. The limiting behavior of positive and negative…
We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established…
We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for…