Erik Burman, Johnny Guzman, Manuel A. Sanchez et al.
We prove an optimal error estimate for the flux variable for a stabilized unfitted Nitsche finite element method applied to an elliptic interface problem with discontinuous constant coefficients. Our result shows explicitly that this error estimate is totally independent of the diffusion coefficients
Johnny Guzman, Michael Neilan
We construct several stable finite element pairs for the Stokes problem on barycentric refinements in arbitrary dimensions. A key feature of the spaces is that the divergence maps the discrete velocity space onto the the discrete pressure space; thus, when applied to models of incompressible flows, the pairs yield divergence-free velocity approximations. The key result is a local inf-sup stability that holds for any dimension and for any polynomial degree. With this result, we construct global divergence-free and stable pairs in arbitrary dimension and for any polynomial degree.
Johnny Guzman, Ridgway Scott
We prove that the Scott-Vogelius finite elements are inf-sup stable on shape-regular meshes for piecewise quartic velocity fields and higher ($k \ge 4$).
Alan Demlow, Johnny Guzman, Alfred H. Schatz
Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global "pollution" term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation. Our chief technical innovation is an improved superapproximation result.
Johnny Guzman, Manuel A. Sanchez, Marcus Sarkis
We define a new finite element method for a steady state elliptic problem with discontinuous diffusion coefficients where the meshes are not aligned with the interface. We prove optimal error estimates in the $L^2$ norm and $H^1$ weighted semi-norm independent of the contrast between the coefficients. Numerical experiments validating our theoretical findings are provided.
Guosheng Fu, Johnny Guzman, Michael Neilan
We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an $n$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.
Johnny Guzmán, Maxim Olshanskii
The paper shows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations which are not fitted to a given smooth or polygonal domain. The property implies stability and optimal error estimates for a class of unfitted finite element methods for the Stokes and Stokes interface problems, such as Nitsche-XFEM or cutFEM. The error analysis is presented for the Stokes problem. All assumptions made in the paper are satisfied once the background mesh is shape-regular and fine enough.
Mark Ainsworth, Johnny Guzmán, Francisco-Javier Sayas
The existence of uniformly bounded discrete extension operators is established for conforming Raviart-Thomas and Nédélec discretisations of $H(div)$ and $H(curl)$ on locally refined partitions of a polyhedral domain into tetrahedra.
Abner J. Salgado, Johnny Guzman, Francisco-Javier Sayas
The analysis of finite-element-like Galerkin discretization techniques for the stationary Stokes problem relies on the so-called LBB condition. In this work we discuss equivalent formulations of the LBB condition.
Johnny Guzman, Ridgway Scott
We prove that an analog of the Scott-Vogelius finite elements are inf-sup stable on certain nondegenerate meshes for piecewise cubic velocity fields. We also characterize the divergence of the velocity space on such meshes. In addition, we show how such a characterization relates to the dimension of C^1 piecewise quartics on the same mesh.
Johnny Guzman, Alexandre Madureira, Marcus Sarkis et al.
We derive error estimates for the piecewise linear finite element approximation of the Laplace--Beltrami operator on a bounded, orientable, $C^3$, surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.
Alexandre Ern, Johnny Guzmán, Pratyush Potu
We construct bounded, commuting projections for the three-dimensional de Rham complex with the additional property that the projections preserve the trace of functions/fields if the latter is a piecewise polynomial in the appropriate trace space. The projections are locally defined and stable in the graph norm. More precisely, the part of the graph norm involving the exterior derivative only involves the oscillation of this derivative in a narrow strip of elements touching the boundary and weighted by the local mesh size. Moreover, the projections are $L^2$-stable locally when acting on functions/fields whose exterior derivative is a piecewise polynomial in the appropriate space. We present two salient applications of the present bounded, commuting, discrete-trace preserving projections: the construction of stable liftings of piecewise polynomial data and an optimality result on the discrete versus continuous extension of piecewise polynomial data.
Johnny Guzmán, Xuehai Huang
Finite element elasticity complexes of low regularity are constructed on tetrahedral Alfeld splits. In comparison with existing three-dimensional elasticity complexes on such splits, the complexes constructed here lower both the Sobolev regularity and the polynomial degrees, while ending in a hybridizable $H({\rm div};\mathbb S)$-conforming symmetric stress space with no vertex degrees of freedom. The construction is obtained from local Bernstein-Gelfand-Gelfand arguments applied to polynomial de Rham complexes on the Alfeld split. Two local polynomial elasticity complexes are proved: an $H^2$-$H^1({\rm inc})$ complex and a lower-regularity $H^1({\rm curl})$-$H({\rm inc}^+)$ complex. Their bubble subcomplexes and dimension formulas are derived. These local exact sequences lead to unisolvent finite elements for the displacement and incompatibility spaces and to global finite element subcomplexes of the corresponding elasticity sequences. In the lowest-order $H^1({\rm curl})$-$H({\rm inc}^+)$ finite element complex, the $H({\rm inc}^+;\mathbb S)$-conforming tensor space is piecewise cubic. At the same order, the terminal stress-displacement pair recovers the Johnson-Mercier-Křížek element, while the construction covers higher-order hybridizable symmetric stresses for all $k\ge1$. A second family gives a low-regularity $H^1$-$H({\rm inc})$ finite element complex for the standard elasticity sequence for all $k\ge2$. Commuting interpolation diagrams are established for both global complexes.
J. Guzman, A. Lischke, M. Neilan
We construct smooth finite elements spaces on Powell-Sabin triangulations that form an exact sequence. The first space of the sequence coincides with the classical $C^1$ Powell-Sabin space, while the others form stable and divergence-free yielding pairs for the Stokes problem. We develop degrees of freedom for these spaces that induce projections that commute with the differential operators.
Johnny Guzman, Manuel A. Sanchez, Marcus Sarkis
We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.