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We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model…
Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research.…
A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a…
We present the construction of a sparse-compressed operator that approximates the solution operator of elliptic PDEs with rough coefficients. To derive the compressed operator, we construct a hierarchical basis of an approximate solution…
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…
We propose a two-level structural optimization method for obtaining an approximate optimal shape of piecewise developable surface without specifying internal boundaries between surface patches. The condition for developability of a…
We consider optimal control problems of elliptic PDEs on hypersurfaces in 2- or 3-dimensional Euclidean space. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral…
In this paper we propose a parallel coordinate descent algorithm for solving smooth convex optimization problems with separable constraints that may arise e.g. in distributed model predictive control (MPC) for linear network systems. Our…
We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present…
For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert…
In this work, we develop a discretisation method for the mixed formulation of the magnetostatic problem supporting arbitrary orders and polyhedral meshes. The method is based on a global discrete de Rham (DDR) sequence, obtained by patching…
A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first order hyperbolic formulation of continuum fluid and solid mechanics. The scheme provably respects the stiff…
If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of…
Differentiable simulators are an emerging concept with applications in several fields, from reinforcement learning to optimal control. Their distinguishing feature is the ability to calculate analytic gradients with respect to the input…
We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net…
Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with…
An algorithm for the computation of global discrete conformal parametrizations with prescribed global holonomy signatures for triangle meshes was recently described in [Campen and Zorin 2017]. In this paper we provide a detailed analysis of…
Polyhedral meshes (PM) - meshes having planar faces - have enjoyed a rise in popularity in recent years due to their importance in architectural and industrial design. However, they are also notoriously difficult to generate and manipulate.…