Volume 33, Issue 4

In this paper, a stabilized space-time finite element scheme on anisotropic quadrilateral meshes for general linear parabolic problems is considered. The scheme is devised on the basis of a unified space-time variational formulation, and uses continuous piece-wise polynomial spaces. The stabilization is achieved by incorporating Streamline-Upwind Petrov–Galerkin (SUPG) techniques. Defining appropriately the stability coefficients, we initially show anisotropic interpolation estimates and then a priori error estimates by following a classical finite element methodology. A series of numerical examples illustrates the theoretical findings.

In this paper, we study adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two-dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.

In Galerkin finite element schemes, the discrete first derivative operator for each spatial dimension is a square matrix that is skew-symmetric under restrictive assumptions for certain types of discretizations and boundary conditions. In most settings, however, this desirable property is violated, often only for a few pairs of nodes. These exceptions can invalidate certain design principles based on the skew-symmetry assumption made for these operators. This paper demonstrates that algebraic manipulations can be performed to make the discrete gradient operators of Bernstein polynomial-based finite element methods skew symmetric. Interest in such discretizations has recently been increasing because they represent natural extensions of second-order algebraic flux correction schemes to higher-order spaces. We employ the new operators in the context of such property-preserving methods, mostly based on discontinuous Galerkin discretizations of arbitrary order. Additional theoretical results for the schemes under investigation are derived, including local and global entropy inequalities, among others. Moreover, a discussion on the optimality of CFL-like time step restrictions arising in explicit Runge–Kutta methods shows that our new approach is superior to earlier representatives of operators employed in similar contexts. These techniques use the monolithic convex limiting paradigm and are applied to the compressible Euler equations.

This paper provides an overview of the new features of the finite element library deal.II , version 9.7.