63 research outputs found
A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators
For the pure biharmonic equation and a biharmonic singular perturbation
problem, a residual-based error estimator is introduced which applies to many
existing nonconforming finite elements. The error estimator involves the local
best-approximation error of the finite element function by piecewise polynomial
functions of the degree determining the expected approximation order, which
need not coincide with the maximal polynomial degree of the element, for
example if bubble functions are used. The error estimator is shown to be
reliable and locally efficient up to this polynomial best-approximation error
and oscillations of the right-hand side
A Mixed Finite Element Method for Singularly Perturbed Fourth Oder Convection-Reaction-Diffusion Problems on Shishkin Mesh
This paper introduces an approach to decoupling singularly perturbed boundary
value problems for fourth-order ordinary differential equations that feature a
small positive parameter multiplying the highest derivative. We
specifically examine Lidstone boundary conditions and demonstrate how to break
down fourth-order differential equations into a system of second-order
problems, with one lacking the parameter and the other featuring
multiplying the highest derivative. To solve this system, we propose a mixed
finite element algorithm and incorporate the Shishkin mesh scheme to capture
the solution near boundary layers. Our solver is both direct and of high
accuracy, with computation time that scales linearly with the number of grid
points. We present numerical results to validate the theoretical results and
the accuracy of our method.Comment: 15 pages, 7 figure
Code generation for generally mapped finite elements
Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations
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Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
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