An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.
Abstract. We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of "finite element Hessian" and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasilinear PDE, all in nonvariational form.
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge-Ampère equation and the Pucci equation.
Abstract. In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework.The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws.In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.1. Introduction. Hyperbolic conservation laws play an important role in many physical and engineering applications. One example is the description of non-viscous compressible flows by the Euler equations. Hyperbolic conservation laws in general only have smooth solutions up to some finite time even for smooth initial data. This makes their analysis and the construction of reliable numerical schemes challenging. The development of discontinuities poses significant challenges to their numerical simulation. Several successful schemes were developed so far and are mainly based on finite differences, finite volume and discontinuous Galerkin (dG) finite element schemes. For an overview on these schemes we refer to [GR96, Krö97, LeV02, Coc03, HW08] and their references. In this work we are interested in a posteriori error control of hyperbolic systems while solutions are still smooth. Our main tools are appropriate reconstructions of the discontinuous Galerkin schemes considered and relative entropy estimates.The first systematic a posteriori analysis for numerical approximations of scalar conservation laws accompanied with corresponding adaptive algorithms, can be traced back to [KO00, GM00], see also [Coc03,DMO07] and their references. These estimates were derived by employing Kruzkov's estimates. A posteriori results for systems were derived in [Laf08, Laf04] for front tracking and Glimm's schemes, see also [KLY10]. For recent a posteriori analysis for well balanced schemes for a damped semilinear wave equation we refer to [AG13].We aim at providing a rigorous a posteriori error estimate for semidiscrete dG schemes applied to systems of hyperbolic conservation laws which are of optimal order. The extension of these results to fully discrete schemes is obviously an important point but exceeds the scope of the work at hand. Our analysis is based on an extension of the reconstruction technique, developed mainly for discretisations of parabolic problems, see [Mak07] and references therein, to space discretisations in the hyperbolic setting. The main idea of the reconstruction technique is to introduce an intermediate function, which we will denote u, which solves a perturbed partial differential equation (PDE). This perturbed PDE is constructed in such a way that this u is sufficiently close to both the approximate solution, denoted u h and the exact solution to the conservation law, denoted u. Then, typically
The fluid-mechanics community is currently divided in assessing the boundaries of applicability of the macroscopic approach to fluid mechanical problems. Can the dynamics of nanodroplets be described by the same macroscopic equations as the ones used for macro-droplets? To the greatest degree, this question should be addressed to the moving contact-line problem. The problem is naturally multiscale, where even using a slip boundary condition results in spurious numerical solutions and transcendental stagnation regions in modelling in the vicinity of the contact line. In this publication, it has been demonstrated via the mutual comparison between macroscopic modelling and molecular dynamics simulations that a small, albeit natural, change in the boundary conditions is all that is necessary to completely regularize the problem and eliminate these nonphysical effects. The limits of macroscopic approach applied to the moving contact-line problem have been tested and validated from the first microscopic principles of molecular dynamic simulations.
In this paper we initiate the study of 2nd order variational problems in L ∞ , seeking to minimise the L ∞ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given H ∈ C 1 (R n×n s ), for the functionalthe associated equation is the fully nonlinear 3rd order PDESpecial cases arise when H is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-Polylaplacian and the ∞-Bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.
We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as recovered finite element methods (R-FEM) has a number of attractive features over both classical finite element and discontinuous Galerkin approaches, most important of which is its potential to produce stable conforming approximations in a variety of settings. Moreover, for special choices of recovery operators, R-FEM produces the same approximate solution as the classical conforming finite element method, while, trivially, one can recast (primal formulation) discontinuous Galerkin methods. A priori error bounds are shown for linear second order boundary value problems, verifying the optimality of the proposed method. Residual-type a posteriori bounds are also derived, highlighting the potential of R-FEM in the context of adaptive computations. Numerical experiments highlight the good approximation properties of the method in practice. A discussion on the potential use of R-FEM in various settings is also included.
Abstract. We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems.
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