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We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

Motivated by the discontinuous Petrov-Galerkin method from Demkowicz & Gopalakrishnan [2011, Numer. Methods Partial Differential Equations, 27, 70-105], we study a variational formulation of second-order elliptic equations in mixed form that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. We apply a Petrov-Galerkin discretization with optimal test functions, or equivalently, minimize the residual in the natural norm associated to the variational form. These optimal test functions can be found by solving local problems. Well-posedness, uniformly in the partition, and optimal error estimates are demonstrated.In the second part of the paper, the application to convection-diffusion problems is studied. The available freedom in the variational formulation and in its optimal Petrov-Galerkin discretization is used to construct a method that allows a (smooth) passing to a converging method in the convective limit, being a necessary condition to retain convergence and having a bound on the cost for a vanishing diffusion. The theoretical findings are illustrated by several numerical results.

Locally supported biorthogonal wavelets are constructed on the unit interval with respect to which second-order constant coefficient differential operators are sparse. As a result, the representation of second-order differential operators on the hypercube with respect to the resulting tensor product wavelet coordinates is again sparse. The advantage of tensor product approximation is that it yields (nearly) dimension-independent rates. An adaptive tensor product wavelet method is applied to solve various singularly perturbed boundary value problems. The numerical results indicate robustness with respect to the singular perturbations. For a two-dimensional model problem this will be supported by theoretical results.

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent iterations. We illustrate our findings by numerical results for the computation of solutions of the Poisson equation with limited Sobolev smoothness on intervals in 1D and L-shaped domains in 2D.

The least-squares mixed finite-element method for second-order elliptic problems yields an approximation u h ∈ V h C H 1 0 (Q) of the potential u together with an approximation p h ∈ r h ⊂ H(div; Q) of the vector field p = -AVu. Comparing u h with the standard finite-element approximation of u in V h , and p h with the mixed finite-element approximation of p, it turns out that they are higher-order perturbations of each other. In other words, they are 'superclose'. Refined a priori bounds and superconvergence results can now be proved. Also, the local mass conservation error is of higher order than could be concluded from the standard a priori analysis.

Superconvergence of the gradient for the linear simplicial finite-element method applied to elliptic equations is a well known feature in one, two, and three space dimensions. In this paper we show that, in fact, there exists an elegant proof of this feature independent of the space dimension. As a result, superconvergence for dimensions four and up is proved simultaneously. The key ingredient will be that we embed the gradients of the continuous piecewise linear functions into a larger space for which we describe an orthonormal basis having some useful symmetry properties. Since gradients and rotations of standard finite-element functions are in fact the rotation-free and divergence-free elements of Raviart-Thomas and Nedelec spaces in three dimensions, we expect our results to have applications also in those contexts.

The extended backward differentiation formulas (EBDFs) and their modified form (MEBDF) were proposed by Cash in the 1980s for solving initial-value problems (IVPs) for stiff systems of ordinary differential equations (ODEs). In a recent performance evaluation of various IVP solvers, including a variable-step-variable-order implementation of the MEBDF method by Cash, it turned out that the MEBDF code often performs more efficiently than codes like RADAU5, DASSL and VODE. This motivated us to look at possible parallel implementations of the MEBDF method. Each MEBDF step essentially consists of successively solving three nonlinear systems by means of modified Newton iteration using the same Jacobian matrix. In a direct implementation of the MEBDF method on a parallel computer system, the only scope for (coarse grain) parallelism consists of a number of parallel vector updates. However, all forward-backward substitutions and all righthand side evaluations have to be done in sequence. In this paper, our starting point is the original (unmodified) EBDF method. As a consequence, two different Jacobian matrices are involved in the modified Newton method, but on a parallel computer system, the effective Jacobian-evaluation and the LU-decomposition costs are not increased. Furthermore, we consider the simultaneous solution, rather than the successive solution, of the three nonlinear systems, so that in each iteration the forward-backward substitutions and the righthand side evaluations can be done concurrently. A mutual comparison of the performance of the parallel EBDF approach and the MEBDF approach shows that we can expect a speedup factor of about 2 on 3 processors.

In this paper we study the discrete approximation of a Dirichlet problem on an interval for a singularly perturbed parabolic PDE. The highest derivative in the equation is multiplied by an arbitrarily small parameter e. If the parameter vanishes, the parabolic equation degenerates to a first-order equation, in which only the time derivative remains. For small values of the parameter, boundary layers may appear that give rise to difficulties when classical discretization methods are applied. Then the error in the approximate solution depends on the value of e. An adapted placement of the nodes is needed to ensure that the error independent of the parameter value and depends only on the number of nodes in the mesh. Special schemes with this property are called e-uniformly convergent. In this paper such e-uniformly convergent schemes are studied, which combine a difference scheme and a mesh selection criterion for the space discretization. Except for a small logarithmic factor, the order of convergence is one and two with respect to the time and space variables, respectively. Therefore, it is of interest to develop methods for which the order of convergence with respect to the time variable is increased. In this paper we develop schemes for which the order of convergence in time can be arbitrarily large if the solution is sufficiently smooth. To obtain uniform convergence, we use a mesh with nodes that are condensed in the neighbourhood of the boundary layers. To obtain a better approximation in time, we use auxiliary discrete problems on the same time-mesh to correct the difference approximations. In this sense, the present algorithm is an improvement over a previously published one. To validate the theoretical results, some numerical results for the new schemes are presented.