In this paper we use the Isogeometric Analysis (IgA) to solve the Helmholtz equation with Dirichlet boundary condition over a bounded physical 2D domain. Starting from the variational formulation of the problem, we show how to apply IgA to obtain an approximated solution based on biquadratic B-spline functions. We focus the attention on problems where the physical domain has very irregular boundary. To solve these problems successfully a high quality parametrization of the domain must be constructed. This parametrization is also a biquadratic tensor product B-spline function, with control points computed as the vertices of a quadrilateral mesh with optimal geometric properties. We study experimentally the influence of the wave number and the parametrization of the physical domain in the accuracy of the approximated solution. A comparison with classical Finite Element Method is also included. The power of IgA is shown solving several difficult model problems, which are particular cases of the Helmholtz equation and where the solution has discontinuous gradient in some points, or it is highly oscillatory. For all model problems we explain how to select the knots of B-spline quadratic functions and how to insert knew knots in order to obtain good approximations. The results obtained with our implementation of the method prove that IgA approach is successful, even on regions with irregular boundary, since it is able to offer smooth solutions having at the same time some singular points and high number of oscillations.

In this paper, we are concerned with an optimal control problem of a quasistatic frictional contact model with history-dependent operators. The contact boundary of the model is divided into two parts where different contact conditions are specified. For the contact problem, we first derive its weak formulation and prove the existence and uniqueness of the solution to the weak formulation. Then we give a priori estimate of the unique solution and prove a continuous dependence result for the solution map. Finally, an optimal control problem that contains boundary and initial condition controls is proposed, and the existence of optimal solutions to the control problem is established.

Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and engineering. The finite element method was used to transform the optimal control problems of an elliptic partial differential equation into a system of linear equations of saddle point form. The main focus of this paper is to characterise and exploit the structure of the coefficient matrix of the saddle point system to build an efficient numerical process. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point problems is a computational task since well known numerical schemes perform poorly if they are not properly preconditioned. The main task of this paper is to construct a preconditioner the mimic the structure of the system coefficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the preconditioned matrix are derived. The main outcome is to achieve optimal convergence results in a small number of iterations with respect to the decreasing mesh size $h$ and the changes in $δ$ the regularisation problem parameters. The numerical results demonstrate the effectiveness and performance of the proposed preconditioner compared to the other existing preconditioners and confirm theoretical results.

This paper provides mathematical and numerical analyses for a dynamic frictionless contact problem in which both of Signorini’s condition and the normal compliance condition are used. The contact problem is considered by employing two viscoelastic (Kelvin-Voigt type) objects: a linear Timoshenko beam and a nonlinear spring. In addition, a transmission condition is imposed on one end of the beam and the top of the spring so that they can touch and vibrate together. We prove the existence of solutions satisfying all the conditions. Time discretizations and finite element methods are utilized to propose the fully discrete numerical schemes. We select several groups of data to present and discuss numerical simulations.

The anisotropic and heterogeneous $N$-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. A recent structure-preserving mixed Galerkin method is applied, leading directly to a finite-dimensional port-Hamiltonian system: its numerical analysis is carried out in a general framework. Optimal choices of mixed finite elements are then proved to reach the best trade-off between the convergence rate and the number of degrees of freedom for the state error. Exta compatibility conditions are identified for the Hamiltonian error to be twice that of the state error, and numerical evidence is provided that some combinations of finite element families meet these conditions. Numerical simulations are performed in 2D to illustrate the main theorems among several choices of classical finite element families. Several test cases are provided, including non-convex domain, anisotropic or heterogeneous cases and absorbing boundary conditions.

In this paper, we present and analyze MINI Mixed finite volume element methods (MINI-FVEM) for Stokes problem on triangular meshes. The trial spaces for velocity and pressure are chosen as MINI element pair, and the test spaces for velocity and pressure are taken as the piecewise constant function spaces on the respective dual grid. It is worth noting that the bilinear form derived from the gradient operator and the bilinear form derived from the divergence are unsymmetric. With the help of two new transformation operators, we establish the equivalence of bilinear forms for gradient operator between finite volume methods and finite element methods, and the equivalence of bilinear forms for divergence operator between finite volume methods and finite element methods, so the inf-sup conditions are obtained. By the element analysis methods, we give the positive definiteness of bilinear form for Laplacian operator. Based on the stability, convergence analysis of schemes are established. Numerical experiments are presented to illustrate the theoretical results.