Second-order Differential Problems Research Articles

Isogeometric discretizations with generalized B-splines: Symbol-based spectral analysis

We perform a spectral analysis of matrices arising from isogeometric discretizations based on hyperbolic and trigonometric generalized B-splines. Second-order differential problems with variable coefficients are considered and discretized by means of sequences of both nested and non-nested generalized spline spaces. We prove that an asymptotic spectral distribution always exists when the matrix-size tends to infinity and is compactly described by a so-called symbol, just as in the polynomial B-spline case. We observe a strong resemblance between the symbol expressions in the hyperbolic, trigonometric and polynomial cases, which results in similar spectral features of the corresponding matrices. The theoretical symbol analysis is illustrated with numerical examples, and we show how the symbol can be used to make an analytical prediction of spectral discretization errors.

Numerical solutions of biharmonic equations on non-convex polygonal domains

To construct C1-continuous basis functions for the numerical solutions of two dimensional biharmonic equations on non-convex domains with clamped and/or simply supported boundary conditions, we use B-spline basis functions instead of conventional Hermite finite element basis functions. The C1-continuous B-spline approximation functions constructed on the master patch are moved onto a physical domain by a geometric patch mapping. However it is difficult to cover a non-convex polygonal domain by one smooth patch mapping. Hence we decompose a non-convex domain into several overlapping rectangular subdomains and use the Schwartz alternating method to assemble local solutions for the global solution. Furthermore, in order to handle the corner singularities arising in non-convex domains, we introduce the implicitly enriched Galerkin method, which is similar to the explicit enrichment techniques used in XFEM, G-FEM, PUFEM, that are successful in handling the singularity problems arising in second-order differential problems.

A version of Krasnosel'skiĭ's compression-expansion fixed point theorem in cones for discontinuous operators with applications

We introduce a new fixed point theorem of Krasnosel’skiĭ type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and discontinuous nonlinearities.

Eigenvibrations of a bar with elastically attached load

We study the problem on the eigenvibrations of a bar with an elastically attached load. The problem is reduced to finding the eigenvalues and eigenfunctions of an ordinary secondorder differential problem with a spectral parameter nonlinearly occurring in the boundary condition at the load attachment point. We prove the existence of countably many simple positive eigenvalues of the differential problem. The problem is approximated by a grid scheme of the finite element method. We study the convergence and accuracy of the approximate solutions.

A high-order coupled compact integrated RBF approximation based domain decomposition algorithm for second-order differential problems

This paper presents a high-order coupled compact integrated RBF (CCIRBF) approximation based domain decomposition (DD) algorithm for the discretisation of second-order differential problems. Several Schwarz DD algorithms, including one-level additive/ multiplicative and two-level additive/ multiplicative/ hybrid, are employed. The CCIRBF based DD algorithms are analysed with different mesh sizes, numbers of subdomains and overlap sizes for Poisson problems. Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here. Especially, the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains, which shows great promise for either serial or parallel computing. For practical tests, we then incorporate the CCIRBF into serial and parallel two-level multiplicative Schwarz. Several numerical examples, including those governed by Poisson and Navier-Stokes equations are analysed to demonstrate the accuracy and efficiency of the serial and parallel algorithms implemented with the CCIRBF. Numerical results show: (i) the CCIRBF-Serial and -Parallel algorithms have the capability to reach almost the same solution accuracy level of the CCIRBF-Single domain, which is ideal in terms of computational calculations; (ii) the CCIRBF-Serial and -Parallel algorithms are highly accurate in comparison with standard finite difference, compact finite difference and some other schemes; (iii) the proposed CCIRBF-Serial and -Parallel algorithms may be used as alternatives to solve large-size problems which the CCIRBF-Single domain may not be able to deal with. The ability of producing stable and highly accurate results of the proposed serial and parallel schemes is believed to be the contribution of the coarse mesh of the two-level domain decomposition and the CCIRBF approximation. It is noted that the focus of this paper is on the derivation of highly accurate serial and parallel algorithms for second-order differential problems. The scope of this work does not cover a thorough analysis of computational time.

A three-point coupled compact integrated RBF scheme for second-order differential problems

In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra information of the nodal first and second derivative values via their identity equations. Owing to its coupling of the information of the nodal first and second derivatives, the CCIRBF scheme becomes more accurate, stable and efficient than the normal compact integrated radial basis function (CIRBF) schemes proposed by [Thai-Quang, N.; Mai-Duy, N.; Tran, C.-D.; Tran-Cong, T. (2012)]. The main features of the CCIRBF scheme include: three-point, high-order accuracy, stability, efficiency and inclusion of boundary values. Several analytic problems are considered to verify the present scheme and to compare its accuracy, stability and efficiency with those of the CIRBF, higher-order compact finite difference (HOC) and some other high-order schemes. Numerical results show that highly accurate and stable results are obtained with the proposed scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the CIRBF and HOC schemes.

A strategy for selecting the frequency in trigonometrically-fitted methods based on the minimization of the local truncation errors and the total energy error

Trigonometrically-fitted methods have been largely used for solving second-order differential problems, and particularly for solving the radial Schrodinger equation (see for instance Alolyan and Simos in J Math Chem 50:782–804, 2012; Simos in J Math Chem 34:39–58, 2003, 44:447–466, 2008; Vigo-Aguiar and Simos in J Math Chem 29:177–189, 2001, 32:257–270, 2002 and the references therein contained). It is well-known that for periodic or oscillatory problems, trigonometrically fitted methods are more efficient than non-fitted methods. A large number of different approaches have been considered in the scientific literature to obtain analytical approximations to the frequency of oscillation in case of periodic solutions, which are valid for a large range of amplitudes of oscillation. However, these techniques have been limited to obtaining only one or two iterates because of the great amount of algebra involved. In this paper we consider the use of a trigonometrically fitted method to obtain numerical approximations for the solutions. This yields very acceptable results provided that the approximation of the parameter of the method is done with great accuracy. Many trigonometrically fitted methods have been reported in the literature, but there is no decisive way to obtain the optimal frequency value. We present a strategy for the choice of the parameter value in the adapted method, based on the minimization of the sum of the total energy error and the local truncation errors in the solution and in the derivative. We include an example solved numerically that confirms the good performance of the strategy adopted.

A compact five-point stencil based on integrated RBFs for 2D second-order differential problems

In this paper, a compact 5-point stencil for the discretisation of second-order partial differential equations (PDEs) in two space dimensions is proposed. We employ integrated radial basis functions in one dimension (1D-IRBFs) to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil. Certain nodal values of the second-order derivatives are incorporated into the approximations with the help of the integration constants. In the case of elliptic PDEs, one algebraic equation is formed at each interior node, and the obtained final system, of which each row has 5 non-zero entries, is solved iteratively using a Picard scheme. In the case of parabolic PDEs discretised with a Crank–Nicolson procedure, a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach. Linear and non-linear test problems, including lid-driven cavity flow and natural convection between the outer square and the inner cylinder, are considered to verify the proposed stencil.

Existence of solutions for second-order differential inclusions involving proximal normal cones

In this work, we prove global existence of solutions for second-order differential problems in a general framework. More precisely, we consider second-order differential inclusions involving proximal normal cone to a set-valued map. This set-valued map is supposed to take admissible values (so in particular uniformly prox-regular values, which may be non-smooth and non-convex). Moreover the solution is required to satisfy an impact law, appearing in the description of mechanical systems with inelastic shocks.

Stability and convergence of mixed discontinuous finite element methods for second-order differential problems

In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.

Stability and convergence of mixed discontinuous finite element methods for second-order differential problems

In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.