Tomas Vejchodsky

Mark Ainsworth, Tomas Vejchodsky

A simple flux reconstruction for finite element solutions of reaction-diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction-diffusion problem. The flux reconstruction is based on simple, independent post-processing operations over patches of elements in conjunction with standard Raviart--Thomas vector fields and gives upper bounds even in cases where Galerkin orthogonality might be violated. If Galerkin orthogonality holds, we prove that the corresponding local error indicators are locally efficient and robust with respect to any mesh size and any size of the reaction coefficient, including the singularly perturbed limit.

Simon L. Cotter, Tomas Vejchodsky, Radek Erban

Stochastic models of chemical systems are often analysed by solving the corresponding Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density.

Shuohao Liao, Tomas Vejchodsky, Radek Erban

Stochastic modelling provides an indispensable tool for understanding how random events at the molecular level influence cellular functions. In practice, the common challenge is to calibrate a large number of model parameters against the experimental data. A related problem is to efficiently study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation). In this paper, tensor-structured parametric analysis (TPA) is presented. It is based on recently proposed low-parametric tensor-structured representations of classical matrices and vectors. This approach enables simultaneous computation of the model properties for all parameter values within a parameter space. This methodology is exemplified to study the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks. The TPA has been implemented in Matlab and the codes are available at http://www.stobifan.org .

Tomas Vejchodsky

We compare three finite element based methods designed for two-sided bounds of eigenvalues of symmetric elliptic second order operators. The first method is known as the Lehmann-Goerisch method. The second method is based on Crouzeix-Raviart nonconforming finite element method. The third one is a combination of generalized Weinstein and Kato bounds with complementarity based estimators. We concisely describe these methods and use them to solve three numerical examples. We compare their accuracy, computational performance, and generality in both the lowest and higher order case.

Tomas Vejchodsky

The standard application of the Lehmann-Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes $σ_i$ that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes $σ_i$ are usually computed by a global saddle point problem solved by mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes $σ_i$ of the same quality. The simplified problem is smaller, it is positive definite, and any $H(\mathrm{div},Ω)$ conforming finite elements, such as Raviart-Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable to compute lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions.

Farid Bozorgnia, Seyyed Abbas Mohammadi, Tomas Vejchodsky

In this paper, we study the first eigenvalue of a nonlinear elliptic system involving $p$-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically and numerically. An alternative proof to show the simplicity of the first eigenvalue is given. In addition, the upper and lower bounds of the first eigenvalue are provided. Then, a numerical algorithm is developed to approximate the principal eigenvalue. This algorithm generates a decreasing sequence of positive numbers and various examples numerically indicate its convergence. Further, the algorithm is generalized to a class of gradient quasilinear elliptic systems.

Ivana Sebestova, Tomas Vejchodsky

We generalize and analyse the method for computing lower bounds of the principal eigenvalue proposed in our previous paper (I. Sebestova, T. Vejchodsky, SIAM J. Numer. Anal. 2014). This method is suitable for symmetric elliptic eigenvalue problems with mixed boundary conditions of Dirichlet, Neumann, and Robin type and it is based on a posteriori error analysis using flux reconstructions. We improve the original result in several aspects. We show how to obtain lower bounds even for higher eigenvalues. We present a local approach for the flux reconstruction enabling efficient implementation. We prove the equivalence of the resulting estimator with the classical residual estimator and consequently its local efficiency. We also prove the convergence of the corresponding adaptive algorithm. Finally, we illustrate the practical performance of the method by numerical examples.

Jana Burkotova, Jitka Machalova, Tomas Vejchodsky

We derive novel guaranteed lower bounds for eigenvalues of the Euler-Bernoulli beam with variable bending stiffness. While the standard finite element Rayleigh-Ritz method automatically yields upper bounds, we obtain lower bounds by employing interpolation error estimates with the explicitly known value of the associated constant. This approach is especially efficient and easy to apply for piecewise constant bending stiffness. For general variable material parameters, we obtain guaranteed lower bounds through an auxiliary beam-bending problem. The first eigenvalue is of primary interest in applications because it represents the critical load that causes buckling of the beam. Our method is, however, suitable also for the higher buckling modes. In addition, it can be applied to the physically more relevant nonlinear Gao beam model with piecewise constant bending stiffness, which has the same first eigenvalue as the classical Euler--Bernoulli beam. The presented numerical experiments illustrate the performance of the proposed eigenvalue bounds, demonstrating their convergence rates.