Nonconforming Virtual Element Method for 2m2m-th Order Partial Differential Equations in Rn\mathbb R^n

A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.Comment: 33page

An adaptive fixed-mesh ALE method for free surface flows

In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.Postprint (published version

An advection-robust Hybrid High-Order method for the Oseen problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $\le k$ on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $\le(k+1)$, a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity-pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element $T$ of diameter $h_T$ contributes to the discretization error with an $\mathcal{O}(h_T^{k+1})$-term in the diffusion-dominated regime, an $\mathcal{O}(h_T^{k+\frac12})$-term in the advection-dominated regime, and scales with intermediate powers of $h_T$ in between. Numerical results complete the exposition

Local projection finite element stabilization for the generalized Stokes problem

We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties. An important feature of the method is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations. Although stability of the method has been established, for the homogeneous Stokes equations, the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space. This, makes it a lot simpler and more attractive. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations

Error analysis of a space-time finite element method for solving PDEs on evolving surfaces

In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations posed on evolving hypersurfaces in $\mathbb{R}^d$, $d=2,3$. The method employs discontinuous piecewise linear in time -- continuous piecewise linear in space finite elements and is based on a space-time weak formulation of a surface PDE problem. Trial and test surface finite element spaces consist of traces of standard volumetric elements on a space-time manifold resulting from the evolution of a surface. We prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm. Furthermore, we derive regularity results for solutions of parabolic PDEs on an evolving surface, which we need in a duality argument used in the proof of the second order convergence estimate