Numerical approximation of a one-dimensional elliptic optimal design problem

We address the numerical approximation by finite-element methods of an optimal design problem for a two phase material in one space dimension. This problem, in the continuous setting, due to high frequency oscillations, often does not have a classical solution, and a relaxed formulation is needed to ensure existence. On the contrary, the discrete versions obtained by numerical approximation have a solution. In this article we prove the convergence of the discretizations and obtain convergence rates. We also show a faster convergence when the relaxed version of the continuous problem is taken into account when building the discretization strategy. In particular it is worth emphasizing that, even when the original problem has a classical solution so that relaxation is not necessary, numerical algorithms converge faster when implemented on the relaxed version.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Rate-independent elastoplasticity at finite strains and its numerical approximation

Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions.   In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations

Analysis of a class of penalty methods for computing singular minimizers

Amongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computa- tional calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalization strategy. We establish convergence results under mild conditions on the stored energy function. Moreover, we present practical strategies for the solution of the discretized problems and for the choice of the penalty parameter

Numerical Methods for Non-divergence Form Second Order Linear Elliptic Partial Differential Equations and Discontinuous Ritz Methods for Problems from the Calculus of Variations

This dissertation consists of three integral parts. Part one studies discontinuous Galerkin approximations of a class of non-divergence form second order linear elliptic PDEs whose coefficients are only continuous. An interior penalty discontinuous Galerkin (IP-DG) method is developed for this class of PDEs. A complete analysis of the proposed IP-DG method is carried out, which includes proving the stability and error estimate in a discrete W2;p-norm [W^2,p-norm]. Part one also studies the convergence of the vanishing moment method for this class of PDEs. The vanishing moment method refers to a PDE technique for approximating these PDEs by a family of fourth order PDEs. Detailed proofs of uniform H1 [H^1] and H2 [H^2]-stability estimates for the approximate solutions and their convergence are presented. Part two studies finite element approximations of a class of calculus of variations problems which exhibit so-called Lavrentiev gap phenomenon (LGP), whose solutions often contain singularities. The LGP incapacitates all standard numerical methods, especially the finite element method, as they fail to produce a correct approximate solution. To overcome the difficulty, an enhanced finite element method based on a truncation technique is developed in this part of the dissertation. The proposed enhanced finite element method is shown to numerically converge on several benchmark problems with the LGP. Part three of the dissertation develops a discontinuous Galerkin numerical framework for general calculus of variations problems, which is called the discontinuous Ritz (DR) methodology and can be regarded as the counterpart of the discontinuous Galerkin (DG) methodology for PDEs. Conceptually, it approximates the admissible space by the DG spaces which consist of totally discontinuous piecewise polynomials and approximates the underlying energy functional by discrete energy functionals defined on the DG spaces. The main idea here is to construct the desired discrete energy functional by using the newly developed DG finite element calculus theory, which only requires replacing the gradient operator in the energy functional by the corresponding DG finite element discrete gradient and adding the standard interior penalty terms. It is shown that for a certain class of functionals the proposed DR method does indeed converge to the true solution