A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi-discrete and a family of fully-discrete time approximate schemes are formulated. These schemes are symmetric. Hp-version error estimates are analyzed for these schemes. For the semi-discrete time scheme a priori L∞(H 1) error estimate is derived and similarly, l∞(H 1) and l2(H 1) for the...

Pointwise a posteriori error estimates are derived for linear secondorder elliptic problems over general polygonal domains in 2D. The analysis carries over regardless of convexity, accounting even for slit domains, and applies to highly graded unstructured meshes as well. A key ingredient is a new asymptotic a priori estimate for regularized Green's functions. The estimators lead always to uppe...

The action of the matrix exponential and related φ functions on vectors plays an important role in the application of exponential integrators to ordinary differential equations. For the efficient evaluation of linear combinations of such actions we consider a new Krylov subspace algorithm. By employing Cauchy’s integral formula an error representation of the numerical approximation is given. Th...

We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rank-one matrices, the new algorithm is fas...

In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the no...

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and mesh quasi-uniform, priori error estimate in L 2 -norm proved. By duality argument Oswald interpolation, posteriori estimates errors W 1, p -seminorm are also obtained. Finally, numerical examples provided to validate ...

In this article we consider a priori error and pointwise estimates for finite element approximations of solutions to semilinear elliptic boundary value problems in d > 2 space dimensions, with nonlinearities satisfying critical growth conditions. It is well-understood how mesh geometry impacts finite element interpolant quality, and leads to the reasonable notion of shape regular simplex meshes...

This paper is concerned with a priori error estimates for the piecewise linear nite element approximation of the classical obstacle problem. We demonstrate by means of two onedimensional counterexamples that the L2-error between the exact solution u and the nite element approximation uh is typically not of order two even if the exact solution is in H 2(Ω) and an estimate of the form ‖u − uh‖H1 ...

In this paper the authors study a nonlinear elliptic-parabolic system, which is motivated by mathematical models for lithium ion batteries. For the reliable and fast numerical solution of the system a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The strategy is justified by an a-priori error estimate for the error between the solution to the coupled system a...