Finite element approximation of source term identification with TV-regularization

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform...

Finite Element Approximation with Hierarchical B-Splines

Positivity preserving finite element approximation

We consider finite element operators defined on “rough” functions in a bounded polyhedron Ω in RN . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of Ω and approximating general functions which do not. We give impossibility results for approximation of general functions to more...

Finite element approximation for time-dependent diffusion with measure-valued source

The convergence of finite element methods for elliptic and parabolic partial differential equations is well-established if source terms are sufficiently smooth. Noting that finite element computation is easily implemented even when the source terms are measure-valued — for instance, modeling point sources by Dirac delta distributions — we prove new convergence order results in two and three dim...

Finite element approximation of the Sobolev constant

Denoting by S the sharp constant in the Sobolev inequality in W 0 (B), being B the unit ball in R, and denoting by Sh its approximation in a suitable finite element space, we show that Sh converges to S as h ↘ 0 with a polynomial rate of convergence. We provide both an upper and a lower bound on the rate of convergence, and present some numerical results.