We present a comprehensive approach to the formulation and discretization of geometric PDE governing processes relevant in biophysics and materials science. We start with key elements of differential geometry and shape differential calculus which enable us to compute first variations of domain and boundary functionals. We propose geometric gradi...
Creator:
Nochetto, Ricardo H. (University of Maryland)
Created:
2013-07-15
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form. Besides the meshsize, a second larger scale is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle (DMP) provided that the mesh is weakly acute. Combining the DMP and weak oper...
Creator:
Nochetto, Ricardo H. (University of Maryland)
Created:
2014-10-22
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
Modeling of thin elastic structures subject to large bending deformations, without much streaching and shearing, leads to (geometrically nonlinear) Kirchhoff plate theories with pointwise isometry constraint. We discuss reduced models for both single and bilayer plates, which give rise to minimization problems for the Hessian of the deformation ...
Creator:
Nochetto, Ricardo H. (University of Maryland)
Created:
2017-06-29
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
In contrast to most of the existing theory of adaptivefiniteelement methods (AFEM), we design an AFEM for -Δ u =f with right hand side f in H -1 instead ofL2. Thishastwo important consequences. First we formulate our AFEM inthenatural space for f, which is nonlocal. Second, we showthatdecay rates for the data estimator are dominated by thosefor...
Creator:
Nochetto, Ricardo H. (University of Maryland)
Created:
2010-12-03
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
Adaptivity is an essential tool in modern scientific andengineering computation that allows one to optimize thecomputational effort by locating the degrees of freedom wherethey are most needed, that is in regions of rapid solutionvariation. Adaptive finite element methods (AFEM) are the mostpopular and effective numerical methods to solve ellipt...
Creator:
Nochetto, Ricardo H. (University of Maryland)
Created:
2010-11-29
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.