A posteriori estimation of the linearization error for strongly monotone nonlinear operators

A-posteriori error estimation for a heterogeneous multiscale method for monotone operators and beyond a periodic setting

In this work we introduce and analyse a heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. The concept of HMM was initially introduced by E and Engquist [E, Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1(1):87–132, 2003]. We first present a macroscopic limit problem for the oscillating non-linear equations and the...

A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems

In this work an efficient approach for a-posteriori error estimation for POD-DEIM reduced nonlinear dynamical systems is introduced. The considered nonlinear systems may also include time and parameter-affine linear terms as well as parametrically dependent inputs and outputs. The reduction process involves a Galerkin projection of the full system and approximation of the system’s nonlinearity ...

A-posteriori Error Analysis for Linearization of Nonlinear Elliptic Problems and Their Discretizations

The paper is devoted to a-posteriori quantitative analysis for errors caused by linearization of nonlinear elliptic boundary value problems and their nite element realizations. We employ duality theory in convex analysis to derive computable bounds on the diierence between the solution of a nonlinear problem and the solution of the linearized problem, by using the solution of the linearized pro...

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then app...

A posteriori error estimation for ellipti eigenproblems