Abstract Multidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

Abstract In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation and led in computational practice to the exclusion of mixed methods with low-order pressure approximations like the Bernardi–Raugel or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly. Thus, inf-sup stable, pressure-robust mixed methods will potentially allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy.

Abstract A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete method is developed and justified for the Sturm–Liouville problem on a finite interval for the Schrödinger equation with a polynomial potential and the boundary conditions of Dirichlet type. The algorithm of the general scheme of our method is developed when the potential function is approximated by the piecewise-constant function. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential polynomial and on the correction number. Our method uses the algebraic operations only and does not need solutions of any boundary value problems and computations of any integrals unlike the previous version. A numerical example illustrates the theoretical results.

Abstract Several finite difference methods are proposed for the infinitesimal generator of 1D asymmetric α-stable Lévy motions, based on the fact that the operator becomes a multiplier in the spectral space. These methods take the general form of a discrete convolution, and the coefficients (or the weights) in the convolution are chosen to approximate the exact multiplier after appropriate transform. The accuracy and the associated advantages/disadvantages are also discussed, providing some guidance on the choice of the right scheme for practical problems, like in the calculation of mean exit time for random processes governed by general asymmetric α-stable motions.

Abstract We present and analyze a space-time Petrov–Galerkin finite element method for a time-fractional diffusion equation involving a Riemann–Liouville fractional derivative of order α ∈ ( 0 , 1 ) {\alpha\in(0,1)} in time and zero initial data. We derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus its well-posedness. Further, we develop a novel finite element formulation, show the well-posedness of the discrete problem, and derive error bounds in both energy and L 2 {L^{2}} norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard L 2 {L^{2}} stability property of the L 2 {L^{2}} projection operator plays a key role. We provide extensive numerical examples to verify the convergence analysis.

Abstract In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ⁢ ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ⁢ ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} (where η ⁢ ( L , N ) , η ⁢ ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ⁢ ( ℓ , N ) ≈ ( η ⁢ ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ⁢ ( Γ ) ⊗ ℒ 2 ⁢ ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ⁢ ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} in random spaces.

Abstract In this short note we analyse a connection between the exponentially graded and a generalisation of the class of S-type meshes for singularly perturbed problems.

Abstract The present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is introduced and the definition of its monotonicity is given. This definition is closely associated with the property of non-negativity of the solution. Under the fulfillment of the positivity condition of the coefficients, two-side estimates of the approximate solution of these vector-difference equations are established and the important a priori estimate in the uniform norm C is given.

Abstract We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L 2 ${L^{2}}$ sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.

Abstract We review the derivation of weakly penalized discontinuous Galerkin methods for scattering dominated radiation transport and extend the asymptotic analysis to non-isotropic scattering. We focus on the influence of the penalty parameter on the edges and derive a new penalty for interior edges and boundary fluxes. We study how the choice of the penalty parameters influences discretization accuracy and solver speed.