Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies on the heterogeneous multiscale method. The time discretization is performed by several classes of Runge-Kutta methods (strongly A-stable or explicit stabilized methods). We discuss the construction and the analysis of such methods for a range of problems, from linear parabolic problems to nonlinear monotone parabolic problems in the very general L ^p (W ^1, p ) setting. We also show that under appropriate assumptions, a computationally attractive linearized method can be constructed for nonlinear problems.

In the present paper we detail the implementation of the Virtual Element Method for two dimensional elliptic equations in primal and mixed form with variable coefficients.

We propose the construction of a class of L ² stable quasi-interpolation operators onto the space of splines on tensor-product meshes, in any space dimension. The estimate we propose is robust with respect to knot repetition and to knot “vicinity” (up to p + 1 knots), so it applies to the most general scenario in which the B-spline functions are known to be well defined.

In this paper we discuss the adjoint stabilised finite element method introduced in Burman (SIAM J Sci Comput 35(6):A2752–A2780, 2013) and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.

In this paper, we review and refine the main ideas for devising the so-called hybridizable discontinuous Galerkin (HDG) methods; we do that in the framework of steady-state diffusion problems. We begin by revisiting the classic techniques of static condensation of continuous finite element methods and that of hybridization of mixed methods, and show that they can be reinterpreted as discrete versions of a characterization of the associated exact solution in terms of solutions of Dirichlet boundary-value problems on each element of the mesh which are then patched together by transmission conditions across interelement boundaries. We then define the HDG methods associated to this characterization as those using discontinuous Galerkin (DG) methods to approximate the local Dirichlet boundary-value problems, and using weak impositions of the transmission conditions. We give simple conditions guaranteeing the existence and uniqueness of their approximate solutions, and show that, by their very construction, the HDG methods are amenable to static condensation. We do this assuming that the diffusivity tensor can be inverted; we also briefly discuss the case in which it cannot. We then show how a different characterization of the exact solution, gives rise to a different way of statically condensing an already known HDG method. We devote the rest of the paper to establishing bridges between the HDG methods and other methods (the old DG methods, the mixed methods, the staggered DG method and the so-called Weak Galerkin method) and to describing recent efforts for the construction of HDG methods (one for systematically obtaining superconvergent methods and another, quite different, which gives rise to optimally convergent methods). We end by providing a few bibliographical notes and by briefly describing ongoing work.

We introduce two robust DPG methods for transient convection-diffusion problems. Once a variational formulation is selected, the choice of test norm critically influences the quality of a particular DPG method. It is desirable that a test norm produce convergence of the solution in a norm equivalent to L ² while producing optimal test functions that can be accurately computed and maintaining good conditioning of the optimal test function solve on highly adaptive meshes. Two such robust norms are introduced and proven to guarantee close to L ² convergence of the primary solution variable. Numerical experiments demonstrate robust convergence of the two methods.

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.

We review basic design principles underpinning the construction of the mimetic finite difference and a few finite volume and finite element schemes for mixed formulations of elliptic problems. For a class of low-order mixed-hybrid schemes, we show connections between these principles and prove that the consistency and stability conditions must lead to a member of the mimetic family of schemes regardless of the selected discretization framework. Finally, we give two examples of using flexibility of the mimetic framework: derivation of arbitrary-order schemes and inexpensive convergent schemes for nonlinear problems with small diffusion coefficients.

This paper reviews the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L ² approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.

In this survey article, we will give a short summary of the basic algorithm issues of discontinuous Galerkin methods for time-dependent convection dominated problems. We will then give a few representative examples of recent developments of discontinuous Galerkin methods for such problems, and provide comparisons with several other types of numerical methods commonly used for similar or related problems. For the comparison, we concentrate mainly on the methods presented in the London Mathematical Society EPSRC Durham Symposium on Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations.

An abstract setting for the construction and analysis of the Multiscale Hybrid-Mixed (MHM for short) method is proposed. We review some of the most recent developments from this standpoint, and establish relationships with the classical lowest-order Raviart-Thomas element and the primal hybrid method, as well as with some recent multiscale methods. We demonstrate the reach of the approach by revisiting the wellposedness and error analysis of the MHM method applied to the Laplace problem. In the process, we devise new theoretical results for this model.