Band 20, Heft 4 - Contains Special Issue Articles Dedicated to the Memory of Academician Alexander Samarskii, Guest Editors: Raytcho Lazarov, Piotr Matus, and Petr Vabishchevich

We consider a class of a generalized time-fractional telegraph equations. The existence of a weak solution of the corresponding initial-boundary value problem has been proved. A finite difference scheme approximating the problem is proposed, and its stability is proved. An estimate for the rate of convergence, in special discrete energetic Sobolev’s norm, is obtained. The theoretical results are confirmed by numerical examples.

We consider the Dirichlet boundary value problem for linear fractional differential equations with the Riemann–Liouville fractional derivatives. By transforming the boundary value problem to the integral equation, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of the boundary value problems by a grid method is discussed and weighted estimates considering the boundary effect are obtained. It is shown that the accuracy (the convergence rate) near the boundary is better than inside the domain due to the influence of the Dirichlet boundary condition.

A class of Lagrangian mixed finite element methods is constructed for an approximate solution of a problem of nonlinear thin elastic shell theory, namely, the problem of finding critical points of the functional of potential energy according to the Budiansky–Sanders model. The proposed numerical method is based on the use of the second derivatives of the deflection as auxiliary variables. Sufficient conditions for the solvability of the corresponding discrete problem are obtained. Accuracy estimates for approximate solutions are established. Iterative methods for solving the corresponding systems of nonlinear equations are proposed and investigated.

The paper is concerned with the reliable numerical solution of a class of nonlinear interface problems governed by the Poisson–Boltzmann equation. Arising in electrostatic biomolecular models these problems typically contain measure-type source terms and their solution often exposes drastically different behaviour in different subdomains. The interface conditions reflect the requirement that the potential and its normal derivative must be continuous. In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this question is not trivial and requires additional analysis, which is based on a special splitting of the problem into simpler subproblems whose weak solutions can be defined in standard Sobolev spaces. This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully computable majorant of the difference between the exact solution of the original problem and any energy-type approximation of it. The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life biophysical systems.

We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.

For Dirichlet initial boundary value problem (IBVP) for two-dimensional quasilinear parabolic equations with mixed derivatives monotone linearized difference scheme is constructed. The ellipticity conditions c 1 ⁢ ∑ α = 1 2 ξ α 2 ≤ ∑ α , β = 1 2 k α ⁢ β ⁢ ( u ) ⁢ ξ α ⁢ ξ β ≤ c 2 ⁢ ∑ α = 1 2 ξ α 2 c_{1}\sum_{\alpha=1}^{2}\xi_{\alpha}^{2}\leq\sum_{\alpha,\beta=1}^{2}k_{\alpha% \beta}(u)\xi_{\alpha}\xi_{\beta}\leq c_{2}\sum_{\alpha=1}^{2}\xi_{\alpha}^{2} are assumed to be fulfilled for the sign alternating solution u ⁢ ( 𝐱 , t ) ∈ D ¯ ⁢ ( u ) {u(\mathbf{x},t)\in\bar{D}(u)} only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle, not only two-sided estimates for the approximate solution y but also its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivatives ∂ ⁡ u ∂ ⁡ x i {\frac{\partial u}{\partial x_{i}}} have discontinuities of the first kind in the neighborhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. The convergence of an approximate solution to a generalized solution of a differential problem in the grid norm L 2 {L_{2}} is proved.

The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x -derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval ( 0 , 1 ] {(0,1]} . For small ε, the problem involves a boundary layer of width 𝒪 ⁢ ( ε ) {\mathcal{O}(\varepsilon)} , where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as { ε ⁢ N } , N 0 → ∞ {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in { ε ⁢ N } {\{\varepsilon N\}} and N 0 {N_{0}} ; as N , N 0 → ∞ {N,N_{0}\to\infty} , the convergence is conditional with respect to N , where N + 1 {N+1} and N 0 + 1 {N_{0}+1} are the numbers of mesh points in x and t , respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x -derivative in the convective term by the central difference operator under the condition h ≤ m ⁢ ε {h\leq m\varepsilon} , which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of 𝒪 ⁢ ( ε - 2 ⁢ N - 2 + N 0 - 1 ) {\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})} . We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for ε = 1 {\varepsilon=1} ), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy in x and first-order accuracy in t .

For a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k , using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k / m {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

Richardson Extrapolation is a very general numerical procedure, which can be applied in the solution of many mathematical problems in an attempt to increase the accuracy of the results. It is assumed that this approach is used to handle non-linear systems of ordinary differential equations (ODEs) which arise often in the mathematical description of scientific and engineering models either directly or after the discretization of the spatial derivatives of partial differential equations (PDEs). The major topic is the analysis of eight advanced implementations of the Richardson Extrapolation. Two important properties are analyzed: (a) the possibility to achieve more accurate results and (b) the possibility to improve the stability properties of eight advanced versions of the Richardson Extrapolation. A two-parameter family of test-examples was constructed and used to check both the accuracy and the absolute stability of the different versions of the Richardson Extrapolation when these versions are applied together with several Explicit Runge–Kutta Methods (ERKMs).

A general superapproximation result is derived in this paper which is useful for the local/interior error analysis of finite element methods.

We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k , of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k m {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form C 0 ⁢ h 2 + C m ⁢ k {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where C m ≤ C ′ + C ′′ m {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}} . This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

We give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived. We prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between T † ⁢ T ⁢ ( X n ) {T^{\dagger}T(X_{n})} and T * ⁢ T ⁢ ( X n ) {T^{*}T(X_{n})} , and the principal angles between X n ∩ ( 𝒩 ⁢ ( T ) ∩ X n ) ⊥ {X_{n}\cap(\mathcal{N}(T)\cap X_{n})^{\perp}} and ( 𝒩 ⁢ ( T ) + X n ) ∩ 𝒩 ⁢ ( T ) ⊥ {(\mathcal{N}(T)+X_{n})\cap\mathcal{N}(T)^{\perp}} . At the end, we consider several specific cases and examples to further illustrate our theorems.

The aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements P k {P_{k}} and P m {P_{m}} ( k < m {k<m} ). Then we analyze the asymptotic relation between these two probabilistic laws when the difference m - k {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is ( 0 , 1 ) d {(0,1)^{d}} for some d ∈ { 1 , 2 , 3 } {d\in\{1,2,3\}} . This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where d = 1 {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in H 1 / 2 ⁢ ( Γ ) {H^{1/2}(\Gamma)} . To avoid computing the latter norm numerically, we realize it using the H 1 ⁢ ( Ω ) {H^{1}(\Omega)} norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the H 1 {H^{1}} and L 2 {L_{2}} norm are proven. We also consider and analyze the case of control constrained problems.