A posteriori error estimates for the monodomain model in cardiac electrophysiology Abstract We consider the monodomain model, a system of a parabolic semilinear reaction–diffusion equation coupled with a nonlinear ordinary differential equation, arising from the (simplified) mathematical description of the electrical activity of the heart. We derive a posteriori error estimators accounting for different sources of error (space/time discretization and linearization). We prove reliability and efficiency (this latter under a suitable assumption) of the error indicators. Finally, numerical experiments assess the validity of the theoretical results. A hybrid iterative algorithm for solving monotone variational inclusion and hierarchical fixed point problems Abstract This paper deals with a strong convergence theorem for a hybrid iterative algorithm without extrapolating step to approximate a common solution of monotone variational inclusion and hierarchical fixed point problems for nonexpansive mappings. Some consequences of the strong convergence theorem are also derived. An application to mixed equilibrium problem is also discussed. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify previously known corresponding results of this area.

On Krylov solutions to infinite-dimensional inverse linear problems Abstract We discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of a given inverse problem, together with a series of model examples and numerical experiments.

Parameterized error bounds for linear complementarity problems of $$B_\pi ^{R}$$ B π R -matrices and their optimal values Abstract New error bounds involving a parameter for linear complementarity problems are presented when the involved matrices are \(B_\pi ^{R}\) -matrices, and the optimal values of these error bounds are determined completely by using the monotonicity of functions of this parameter. It is shown that the optimal error bounds are sharper than that provided by García-Esnaola and Peña (Calcolo 54(3):813–822, 2017) under certain assumptions.

Adaptive fixed point iterations for semilinear elliptic partial differential equations Abstract In this paper we study the behaviour of finite dimensional fixed point iterations, induced by discretization of a continuous fixed point iteration defined within a Banach space setting. We show that the difference between the discrete sequence and its continuous analogue can be bounded in terms depending on the discretization of the infinite dimensional space and the contraction factor, defined by the continuous iteration. Furthermore, we show that the comparison between the finite dimensional and the continuous fixed point iteration naturally paves the way towards a general a posteriori error analysis that can be used within the framework of a fully adaptive solution procedure. In order to demonstrate our approach, we use the Galerkin approximation of singularly perturbed semilinear monotone problems. Our scheme combines the fixed point iteration with an adaptive finite element discretization procedure (based on a robust a posteriori error analysis), thereby leading to a fully adaptive fixed-point-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach.

Common solutions to a finite family of inclusion problems and an infinite family of fixed point problems by a generalized viscosity implicit scheme including applications Abstract This manuscript deals with two problems: the first one is a variational inclusion problem involving an m-accretive mapping and a finite family of inverse strongly accretive mappings, and the other one is a fixed point problem having an infinite family of strict pseudo-contraction mappings in Banach spaces. To approximate the common solution of these problems, we design a generalized viscosity implicit iterative scheme with Meir–Keeler contraction. A strong convergence result for the proposed iterative scheme is established. Applications based on convex minimization problem, linear inverse problem, variational inequality problem and equilibrium problem are derived from the main result. The numerical applicability of the main result and some applications are demonstrated by three examples. Our result extends, generalizes and unifies the previously known results given in literature.

A Riemann solution approximation based on the zero diffusion–dispersion limit of Dafermos reformulation type problem Abstract In the present work, a new numerical strategy is designed to approximate the Riemann solutions of systems of conservation laws. Here, the main difficulty comes from the definition of the discontinuous solutions. Indeed, the shock solutions are no longer selected by entropy criterion but they are defined as the zero limit of a diffusive–dispersive system. As a consequence, the solutions of interest may contain non classical shocks. In order to derive a suitable numerical approach, the Dafermos diffusion technique is adopted here. Then, the PDE initial value problem is reformulated as an ODE boundary value problem. A fourth-order finite difference scheme is introduced to approximate the solution of this ODE boundary value problem. In this work, a particular attention is paid on the existence of discrete solutions and several numerical experiments illustrate the relevance of the derived numerical strategy.

Convergence rate of the finite element approximation for extremizers of Sobolev inequalities on 2D convex domains Abstract We investigate a FEM-based numerical scheme approximating extremal functions of the Sobolev inequalities. The main result of this paper shows that if the domain is polygonal and convex in \(\mathbb {R}^2\), then the convergence rate of a finite element solution to an exact extremal function is \(O(h^2)\) in the \(L^2\) norm, and it is O(h) in the \(H^1\) norm, where h denotes the mesh size of a triangulation of the domain.

Alternating positive semidefinite splitting preconditioners for double saddle point problems Abstract In this paper, an alternating positive semidefinite splitting (APSS) preconditioner is proposed for solving double saddle point problems. The corresponding APSS iteration method is proved to be convergent unconditionally. Moreover, to further improve its efficiency, a relaxed variant is established for the APSS preconditioner, which results in better spectral distribution and numerical performance. Numerical experiments with liquid crystal director models demonstrate the effectiveness of the APSS preconditioner and its relaxed variant when compared with other preconditioners. Comparison between the related numerical results shows that the proposed preconditioners are comparable with (though not really better than) the best existing preconditioners.

Semi-implicit Milstein approximation scheme for non-colliding particle systems Abstract We introduce a semi-implicit Milstein approximation scheme for some classes of non-colliding particle systems modeled by systems of stochastic differential equations with non-constant diffusion coefficients. We show that the scheme converges at the rate of order 1 in the mean-square sense.