Sergey Korotov

On Nonobtuse Simplicial Partitions

This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

On discrete maximum principles for nonlinear elliptic problems

In order to have reliable numerical simulations it is very important to preserve basic qualitative properties of solutions of mathematical models by computed approximations. For scalar second-order elliptic equations, one of such properties is the maximum principle. In our work, we give a short review of the most important results devoted to discrete counterparts of the maximum principle (called discrete maximum principles, DMPs), mainly in the framework of the finite element method, and also present our own recent results on DMPs for a class of second-order nonlinear elliptic problems with mixed boundary conditions.

An Algebraic Discrete Maximum Principle in Hilbert Space with Applications to Nonlinear Cooperative Elliptic Systems

Discrete maximum principles are derived for finite element discretizations of nonlinear elliptic systyems with cooperative and weakly diagonally dominant coupling. The results are achieved via an algebraic discrete maximum principle organized in a Hilbert space framework and, in the case of simplicial elements, are obtained under weakened acute type conditions for the finite element (FE) meshes.

On the equivalence of ball conditions for simplicial finite elements in R-d

We prove that the inscribed and circumscribed ball conditions, commonly used in finite element analysis, are equivalent in any dimension.

Strong regularity of a family of face-to-face partitions generated by the longest-edge bisection algorithm

We examine the longest-edge bisection algorithm that chooses for bisection the longest edge in a given face-to-face simplicial partition of a bounded polytopic domain in ℝd. Dividing this edge at its midpoint, we define a locally refined partition of all simplices that surround this edge. Repeating this process, we obtain a family ℱ = {ℐh}h → 0 of nested face-to-face partitions ℐh. For d = 2, we prove that this family is strongly regular; i.e., there exists a constant C > 0 such that meas T ≥ Ch2 for all triangles T ∈ ℐh and all triangulations ℐh ∈ ℱ. In particular, the well-known minimum angle condition is valid.

On Weakening Conditions for Discrete Maximum Principles for Linear Finite Element Schemes

In this work we discuss weakening requirements on the set of sufficient conditions due to Ph. Ciarlet [4,5] for matrices associated to linear finite element schemes, which is commonly used for proving validity of discrete maximum principles (DMPs) for the second order elliptic problems.

Goal-oriented a posteriori error estimates for transport problems

Some aspects of goal-oriented a posteriori error estimation are addressed in the context of steady convection-diffusion equations. The difference between the exact and approximate values of a linear target functional is expressed in terms of integrals that depend on the solutions to the primal and dual problems. Gradient averaging techniques are employed to separate the element residual and diffusive flux errors without introducing jump terms. The dual solution is computed numerically and interpolated using higher-order basis functions. A node-based approach to localization of global errors in the quantities of interest is pursued. A possible violation of Galerkin orthogonality is taken into account. Numerical experiments are performed for centered and upwind-biased approximations of a 1D boundary value problem.

Discrete maximum principle for parabolic problems solved by prismatic finite elements

In this paper we analyze the discrete maximum principle (DMP) for a non-stationary diffusion-reaction problem solved by means of prismatic finite elements and @q-method. We derive geometric conditions on the shape parameters of prismatic partitions and time-steps which a priori guarantee validity of the DMP. The presented numerical tests illustrate the sharpness of the obtained conditions.

Error control in terms of linear functionals based on gradient averaging techniques

We show how commonly used gradient averaging techniques can be successfully applied to estimation of computational errors evaluated by linear (goal-oriented) functionals for linear elliptic type problems. General scheme for construction of corresponding estimators is described and effectivity of the proposed approach is demonstrated in numerical tests.

A Comparison of Simplicial and Block Finite Elements

In this note we discuss and compare the performance of the finite element method (FEM) on two popular types of meshes – simplicial and block ones. A special emphasis is put on the validity of discrete maximum principles and on associated (geometric) mesh generation/refinement issues in higher dimensions. As a result, we would recommend to carefully reconsider the common belief that the simplicial finite elements are very convenient to describe complicated geometries (which appear in real-life problems), and also that the block finite elements, due to their simplicity, should be used if the geometry of the solution domain allows that.