In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal...

In this paper, a priori error estimates for space-time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from [23, 24], where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuou...

For a simply-connected Lipschitz domain Ω ⊂ lR d and m ∈ lN 0 we consider the quotient space W m+1,p (Ω)/P m (Ω), p ∈ [1, ∞], whose elements are equivalence classes [u] according to [u] := { w ∈ W m+1,p (Ω) | w − u ∈ P m (Ω) }. We recall that W m+1,p (Ω)/P m (Ω) is a Banach space with respect to the Sobolev quotient norm (4.1) [u] m+1,p,Ω := inf p∈P m (Ω) u + p m+1,p,Ω. A basic result states th...

We present constructive a priori error estimates forH 0 -projection into a space of polynomials on a one-dimensional interval. Here, “constructive” indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form. Using the properties of Legendre polynomials, we consider a method by which to determine these constants t...

In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and DG formulation in these methods allows local mes...

Given a C-function f on a compact riemannian manifold (X, g) we give a set of frequencies L = Lf (ε) depending on a small parameter ε > 0 such that the relative L-error ‖f−f ‖ ‖f‖ is bounded above by ε, where f L denotes the L-partial sum of the Fourier series f with respect to an orthonormal basis of L(X) constituted by eigenfunctions of the Laplacian operator ∆ associated to the metric g.

In this work, we show quasi-optimal interface error estimates for solutions obtained by the symmetric interior penalty discontinuous Galerkin method. It is proved that the numerical solution restricted to an interface converges with order |lnh|h under suitable regularity requirements, where the logarithmic factor is only present in the lowest order case, i.e., k = 1. For this case, we also deri...

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L2 and the H1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficien...