A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

We design and analyze a $$C^0$$ C 0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions.

p- and hp- virtual elements for the Stokes problem

We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.

On superconvergence of Runge–Kutta convolution quadrature for the wave equation

The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge–Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $$\left| s\right| $$ s (up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.

Space–time discontinuous Galerkin approximation of acoustic waves with point singularities

We develop a convergence theory of space–time discretizations for the linear, second-order wave equation in polygonal domains $\varOmega \subset{\mathbb R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space–time DG formulation developed in Moiola & Perugia (2018, A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math., 138, 389–435), we (a) prove optimal convergence rates for the space–time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable sparse space–time version of the DG scheme. The latter scheme is based on the so-called combination formula, in conjunction with a family of anisotropic space–time DG discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\varOmega $ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space–time DG schemes.

Matrix equation solving of PDEs in polygonal domains using conformal mappings

We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz-Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation.

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L2 penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.