Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4th order eigenvalue problems with variable coefficients

We apply the mixed Petrov–Galerkin formulation to construct finite element approximations for transient and steady-state creep problems. With the new approach we recover stability, convergence, and accuracy of some Galerkin unstable approximations. We also present the main results on the numerical analysis and error estimates of the proposed finite element approximation for the steady problem, and discuss the asymptotic behavior of the continuum and discrete transient problems.

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Mixed finite element approximation of a degenerate elliptic problem

Numerische Mathematik ◽

10.1007/s002110050142 ◽

1995 ◽

Vol 71 (2) ◽

pp. 225-236 ◽

Cited By ~ 7

Author(s):

Donatella Marini ◽

Paola Pietra

Keyword(s):

Finite Element ◽

Elliptic Problem ◽

Mixed Finite Element ◽

Finite Element Approximation ◽

Element Approximation ◽

Degenerate Elliptic

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A mixed finite-element method for elliptic operators with Wentzell boundary condition

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry068 ◽

2018 ◽

Vol 40 (1) ◽

pp. 87-108

Author(s):

Eberhard Bänsch ◽

Markus Gahn

Keyword(s):

Boundary Condition ◽

Finite Element ◽

Mixed Finite Element ◽

A Priori ◽

Mixed Finite Element Method ◽

Finite Element Approximation ◽

Element Approximation ◽

Weak Formulation ◽

Wentzell Boundary Condition ◽

Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

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Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202519500295 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1585-1617 ◽

Cited By ~ 2

Author(s):

Yvon Maday ◽

Carlo Marcati

Keyword(s):

Finite Element ◽

Discontinuous Galerkin ◽

Sobolev Spaces ◽

Eigenvalue Problems ◽

Weighted Sobolev Spaces ◽

Finite Element Approximation ◽

Three Dimensions ◽

Element Approximation ◽

Singular Potentials ◽

Dg Method

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.

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