Related papers: Biharmonic obstacle problem: guaranteed and comput…
The paper is concerned with a class of nonlinear free boundary problems, which are usually solved by variational methods based on primal (or primal-dual) variational settings. We deduce and investigate special relations (error identities).…
This paper is concerned with the two--phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of Repin and Valdman (2015) and verify them numerically on two examples in two space dimensions. A…
For a class of variational problems with linear differential operator, we obtain a convenient form of the deviation identity, i.e., the value of the distance between approximated solutions and the exact ones. We illustrate the result with…
We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution…
Motivated by problems in contact mechanics, we propose a duality approach for computing approximations and associated a posteriori error bounds to solutions of variational inequalities of the first kind. The proposed approach improves upon…
We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm and a certain nonlinear…
This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence…
We verify functional a posteriori error estimate proposed by S. Repin for a class of obstacle problems. The obstacle problem is formulated as a quadratic minimization problem with constrains equivalently formulated as a variational…
We optimize the running time of the primal-dual algorithms by optimizing their stopping criteria for solving convex optimization problems under affine equality constraints, which means terminating the algorithm earlier with fewer…
This paper demonstrates a practical method for computing the solution of an expectation-constrained robust maximization problem with immediate applications to model-free no-arbitrage bounds and super-replication values for many financial…
A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems,…
In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of…
We apply duality theory to discretized convex minimization problems to obtain computable guaranteed upper bounds for the distance of given discrete functions and the exact discrete minimizer. Furthermore, we show that the discrete duality…
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…
This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in…
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…
For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…
We prove an optimal order error bound in the discrete $H^2(\Omega)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose…
Primal-dual interior-point methods solve constrained convex optimization problems to tight tolerances with speed and robustness. Their solutions are also efficiently differentiable with respect to the problem data through the implicit…
We propose a new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous…