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The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. [ABSTRACT FROM AUTHOR]

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when k$$ k $$ is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate. [ABSTRACT FROM AUTHOR]

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → R + is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the L 2 norm. A series of numerical experiments are conducted to validate the theoretical developments. [ABSTRACT FROM AUTHOR]

We investigate some nonlinear elliptic problems of the form \Delta^2v + \sigma(x) v= h(x,v)\quadin \Omega,\quad v=\Delta v=0 \quadon \partial\Omega, \eqno({\rm P}) where $\Omega$ is a regular bounded domain in $\mathbb{R}^N$, $N\geq2$, $\sigma(x)$ a positive function in $L^{\infty}(\Omega)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

The paper investigates the approximate properties of biharmonic Poisson integrals for the upper half-plane on the classes of Hölder functions in the uniform metric. The exact values for the upper bounds of the deviations of the functions of the Hölder class H1 from the biharmonic Poisson integrals defined in the upper half-plane were found. [ABSTRACT FROM AUTHOR]

We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic Choquard-type problem Δ 2 u - β Δ u = λ u + (I μ ∗ F (u)) f (u) , in R 4 , ∫ R 4 | u | 2 d x = c 2 > 0 , u ∈ H 2 (R 4) , where β ≥ 0 , λ ∈ R , I μ = 1 | x | μ with μ ∈ (0 , 4) , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem. [ABSTRACT FROM AUTHOR]

We investigate some nonlinear elliptic problems of the form (P) ∆2 v + σ(x)v = h(x, v) in Ω, v = ∆v = 0 on ∂Ω, where Ω is a regular bounded domain in RN, N > 2, σ(x) a positive function in L∞(Ω), and the nonlinearity h(x, t) is indefinite. We prove the existence of solutions to the problem (P) when the function h(x, t) is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical. [ABSTRACT FROM AUTHOR]

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen [SIAM J. Sci. Comput., 44 (2022), pp. A1205-A1226] recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polyg- onal domains. In this paper, we present the LARS algorithm (lightning-AAA rational Stokes) for computing two-dimensional (2D) Stokes flows in domains with smooth boundaries and multiply con- nected domains using lightning and AAA rational approximation [Y. Nakatsukasa, O. Sète, and L. N. Trefethen, SIAM J. Sci. Comput., 40 (2018), pp. A1494-A1522]. After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error. [ABSTRACT FROM AUTHOR]

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems. [ABSTRACT FROM AUTHOR]

We consider the biharmonic equation Δ²u-(a + b R R5 |Δu|²dx) Δu + V (x)u = f(u), where V (x) and f(u) are continuous functions. By using a perturbation approach and the symmetric mountain pass theorem, the existence and multiplicity of solutions for this equation are obtained, and the power-type case f(u) = |u|p-2u is extended to p (2, 10), where it was assumed p (4, 10) in many papers. [ABSTRACT FROM AUTHOR]

In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the existence of radial solutions for a class of biharmonic equation with a convection term, involving two real parameters A and p. We mainly use a combination of the fixed point index theory and the Banach contraction theorem to prove that there are A0 > 0 and p0 > 0 such the equation admits at least one radial solution for all (A, p) e [-Ao, x [0, p0]. [ABSTRACT FROM AUTHOR]

The primary objective of this article is to consider a class of biharmonic equations with Navier boundary conditions. Among others, a new criterion for the existence of positive solution is established. In addition, some new uniqueness and multiplicity results of positive radial solution are also considered. [ABSTRACT FROM AUTHOR]

The manifolds with singularities are of great importance in geometric analysis. In general, the singularities include cones, edges, corners, or higher order singularities. In this paper, we study the following Dirichlet problem for a class of degenerate fourth-order elliptic equations on the conical singular manifolds. Δ B 2 u = λ u + | u | p - 2 u in int B , u = ∂ u ∂ ν = 0 on ∂ B , where λ ≥ 0 , 2 < p < 2 ∗ = 2 n n - 4 , ν is the unit outward normal vector to the boundary of B . 2 ∗ = 2 n n - 4 is the critical cone Sobolev exponent for the fourth-order problem and n ≥ 5 . We first introduce the background of conical manifolds and the weighted cone Sobolev spaces. Then we recall the cone Sobolev inequality and cone Poincaré inequality. With the help of variational method, we obtain the existence of non-trivial weak solutions in the weighted cone Sobolev space H 2 , 0 2 , n 2 (B) . [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵2u=f(u,▵u) on an annular domain Ω in RN with the Navier boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:R+×R−→R+ is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f(ξ,η) to have superlinear or sublinear growth on ξ,η as |(ξ,η)|→0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones.

New representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found.

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∊ V := H2 0 (\Omega) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces Vh(P) and a smoother allows rough source terms F ∊ V* =H-20(Ω). The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator J: Vh → V from the nonconforming virtual element space Vh. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on u ∊ V. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement. [ABSTRACT FROM AUTHOR]

In this paper, the following critical biharmonic elliptic problem { Δ 2 u = λ u + μ u ln ⁡ u 2 + | u | 2 ⁎ ⁎ − 2 u , x ∈ Ω , u = ∂ u ∂ ν = 0 , x ∈ ∂ Ω is considered, where Ω ⊂ R N is a bounded smooth domain with N ≥ 5. Some interesting phenomena occur due to the uncertainty on the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at least one nontrivial weak solution under some appropriate assumptions of λ and μ. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced. [ABSTRACT FROM AUTHOR]

In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing H^2-conforming virtual element, an H^1-nonconforming virtual element is obtained with the same degrees of freedom as the usual H^1-conforming virtual element, such that it locally has H^2-regularity on each polygon in meshes. To enforce the C^1 continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem. [ABSTRACT FROM AUTHOR]

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré–Korn-type inequalities for hybrid fields.

In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx + 2uxxyy + uyyyy = f(u). To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: Δ (Δ f) = g , where g : D ¯ → C is a continuous function and D ¯ denotes the closure of the unit disk D in the complex plane C . In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality 1 - | z | 2 1 - | f (z) | 2 ≤ C , where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric. [ABSTRACT FROM AUTHOR]

In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A |$C^0$| finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new mixed method proposed by Rafetseder and Zulehner for Kirchhoff plates and apply it to fourth order eigenvalue problems. Using two auxiliary variables this new mixed method makes it possible to require only H 1 regularity for the displacement and the auxiliary variables, without the demand of a convex domain. We provide a direct comparison, specifically in view of convergence orders, to the C 0 -IPG method and Ciarlet-Raviart's mixed method of vibration problems with the boundary conditions of the clamped plate and the simply supported plate. The numerical experiments are done with the open-source finite element library deal.II and include the implementation of the coupling of finite elements with elements on the boundary to incorporate non-trivial boundary conditions. [ABSTRACT FROM AUTHOR]

The main objective of this article is to consider a biharmonic problem with Navier boundary conditions. Among others, some criteria for the existence, multiplicity and nonexistence of positive solutions are established by employed fixed point theorems in a cone. In addition, we not only consider the sublinear case, but also we will study the case of appropriate combinations of superlinearity and sublinearity. [ABSTRACT FROM AUTHOR]

A new algorithm for the numerical solution of the biharmonic equation is developed. It is based on the first implemented hp-version of the least-squares collocation method (hp-LSCM) with integral collocations for a fourth-order elliptic equation in combination with modern methods of accelerating iterative processes for solving systems of linear algebraic equations (SLAE). The hp-LSCM provides the possibilities to refine the grid (h-version) and increase the degree of polynomials to the arbitrary order (p-approach). The convergence of approximate solutions obtained by the implemented version of the method is analyzed using an example of a numerical simulation of the bending of a hinged isotropic plate. The high accuracy and the increased order of convergence using polynomials up to the tenth order in the hp-LSCM are shown. The effectiveness of the combined application of algorithms for accelerating iterative processes to solve SLAE that are combined with LSCM is investigated. In this paper, we consider the application of the following algorithms: preconditioning of SLAE matrices; the iteration acceleration algorithm based on Krylov subspaces; the prolongation operation on a multigrid complex; parallelization using OpenMP; a modified algorithm for solving local SLAEs. The latter is implemented with iterations over subdomains (which are cells) and makes it possible to more effectively solve overdetermined SLAEs in the LSCM in the case of solving a linear differential equation. The form of the matrices does not change at each iteration. Only the elements of the vectors of their right parts corresponding to the matching conditions are modified. The calculation time on a personal computer is reduced by more than 350 times with the combined use of all acceleration techniques compared to the case when only preconditioning was used.

Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational polynomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.

In this paper, we present an enriched Ciarlet-Raviart scheme for the biharmonic equation with variable coefficient on Lipschitz (maybe nonconvex)polyhedral domains. With the enriched finite element space for the Laplacian of the true solution, we manage to prove the discrete $ H^{2} $-stability of numerical solution. Error analysis is provided for solutions with low regularity. [ABSTRACT FROM AUTHOR]

In this paper, we study a singular nonhomogenous biharmonic problem with Dirichlet boundary condition in Heisenberg group$ \begin{equation*} \left \{\begin{array}{ll} \Delta^2_{{\mathbb{H}^1}}u = \frac{f(\xi, u)}{\rho(\xi)^\beta}+\epsilon h(\xi)\ &\mbox{in} \ \Omega, \\ u = \frac{\partial u}{\partial \nu} = 0, \ \ &\mbox{on}\ \partial\Omega, \end{array} \right. \end{equation*} $where $ \Omega\subset {\mathbb{H}^1} $ is a bounded smooth domain, $ \Delta^2_{{\mathbb{H}^1}}u = \Delta_{\mathbb{H}^1}(\Delta_{\mathbb{H}^1}u) $ denotes the biharmonic operator in Heisenberg group $ \mathbb{H}^1 = \mathbb{C} \times \mathbb{R} $, $ 0\leq \beta < 4 $ with $ 4 $ is the homogeneous dimension of $ {\mathbb{H}^1} $ and $ f:\Omega \times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function which satisfies subcritical and critical exponential growth condition, $ h(\xi)\in (D_0^{2, 2}(\Omega))^* $, $ h(\xi)\geq0 $ and $ h(\xi) \not\equiv0 $, $ \rho(\xi) = (|z|^4+t^2)^{\frac{1}{4}} $, $ \xi = (z, t)\in \mathbb{H}^1 $ with $ z = (x, y)\in\mathbb{R}^{2} $, $ \epsilon $ is a small positive parameter. We obtain the existence and multiplicity of solutions by the Ekeland variational principle, mountain pass theorem and singular Adams inequality in Heisenberg group. [ABSTRACT FROM AUTHOR]

In this work we develop a novel fully discrete version of the plates complex, an exact Hilbert complex relevant for the mixed formulation of fourth-order problems. The derivation of the discrete complex follows the discrete de Rham paradigm, leading to an arbitrary-order construction that applies to meshes composed of general polygonal elements. The discrete plates complex is then used to derive a novel numerical scheme for Kirchhoff–Love plates, for which a full stability and convergence analysis are performed. Extensive numerical tests complete the exposition. [ABSTRACT FROM AUTHOR]

In this paper, we are inspired by Ngô, Nguyen and Phan's (2018 Nonlinearity 31 5484–99) study of the pointwise inequality for positive C 4-solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality − Δ u u + α | ∇ u | 2 u 2 + β u − q + 1 2 ⩽ C R 2 for positive C 4-solutions of the biharmonic equations with negative exponent − Δ 2 u = u − q i n B R where B R denotes the ball centered at x 0 with radius R, n ⩾ 3, q > 1, and some constants α ⩾ 0, β ⩾ 0, C > 0. [ABSTRACT FROM AUTHOR]

New representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found [ABSTRACT FROM AUTHOR]

This paper studies a nonlocal biharmonic evolution equation with Dirichlet boundary condition that arises in image restoration. We prove the existence and uniqueness of solutions to the nonlocal problem by the variational method and show that the solutions of the nonlocal problem converge to the solution of the classical biharmonic equation with Dirichlet boundary condition if the nonlocal kernel is rescaled appropriately. The asymptotic behavior is discussed. Besides, we study the Navier problem by transforming it into a Dirichlet problem with a fixed point. The existence, uniqueness, convergence under the rescaling of the kernel, and asymptotic behavior of solutions to the Navier problem are discussed. [ABSTRACT FROM AUTHOR]

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]

Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson's problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases. [ABSTRACT FROM AUTHOR]

A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves fast Fourier transforms and, in comparison to its one-step counterpart, it has the advantage of requiring the solution a symmetric positive definite Schur complement system rather than a nonsymmetric one. As a consequence, the corresponding step of the new method is performed using a preconditioned conjugate gradient method. The total cost of the method on a N × N partition of the unit square is O (N 2 log N) . To demonstrate the optimal accuracy of the method, the results of numerical experiments are provided. [ABSTRACT FROM AUTHOR]