Related papers: Uniform error estimates for general semilinear tur…
This work studies the total variation regularized $\ell_2$ estimator (fused lasso) in the setting of a change point detection problem. Compared with existing works that focus on the sum of squared estimation errors, we give bound on the…
We study adaptive mesh selection for the solution of systems of initial value problems. The goal is a rigorous theoretical analysis of potential advantages of adaption. For an optimal method in the sense of the speed of convergence, we…
An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework…
This work reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations,…
A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis.…
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the…
This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy…
We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild…
The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The…
We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager--Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the…
We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…
In this note we examine the a priori and a posteriori analysis of discontinuous Galerkin finite element discretisations of semilinear elliptic PDEs with polynomial nonlinearity. We show that optimal a priori error bounds in the energy norm…
There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes…
Singular degenerate differential operator equations are studied. The uniform separability of boundary value problems for degenerate elliptic equation and optimal regularity properties of Cauchy problem for degenerate parabolic equation are…
We derive a new residual-type a posteriori estimator for a singularly perturbed reaction-diffusion problem with obstacle constraints. It generalizes robust residual estimators for unconstrained singularly perturbed equations. Upper and…
In this paper, we investigate optimal control problems governed by the parabolic interface equation, in which the control acts on the interface. The solution to this problem exhibits low global regularity due to the jump of the coefficient…
We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of…
We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point…
We show that a certain error estimate for a fully discrete finite element approximation of the solution of the heat equation which is defined in a two-dimensional Euclidean domain carries over to the case of a general linear parabolic…
We obtain sharp convergence rates, using Dirichlet correctors, for solutions of wave equations in a bounded domain with rapidly oscillating periodic coefficients. The results are used to prove the exact boundary controllability that is…