Strongly elliptic pseudodifferential equations on the sphere with radial basis functions

Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and collocation methods. A salient feature of the paper is a unified theory for error analysis of both approximation methods.     

Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.     

Adjusted Sparse Tensor Product Spectral Galerkin Method for Solving Pseudodifferential Equations on the Sphere with Random Input Data

An adjusted sparse tensor product spectral Galerkin approximation method based on spherical harmonics is introduced and analyzed for solving pseudodifferential equations on the sphere with random input data. These equations arise from geodesy where the sphere is taken as a model of the earth. Numerical solutions to the corresponding \(k\)-th order statistical moment equations are found in adjusted sparse tensor approximation spaces which are accordingly designed to the regularity of the data and the equation. Established convergence theorem shows that the adjusted sparse tensor Galerkin discretization is superior not only to the full tensor product but also to the standard sparse tensor counterpart when the statistical moments of the data are of mixed unequal regularity. Numerical experiments illustrate our theoretical results.

     

On the numerical construction of an attractor for the Navier-Stokes system

In this paper the problem of the numerical approximation of the minimal global B-attractor for a semiflow generated by the Navier-Stokes equations in a two-dimensional bounded domain Ω is considered. The method suggested here is based on the formula , where G^N is a sequence of compact subsets of L₂(Ω), . The procedure of constructing G^N is finite and includes the numerical solution of the Navier-Stokes equations by means of the Galerkin method, together with an explicit finite-difference discretization in time. Bibliography: 5 titles. To dear teacher Olga A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 91–97. Translated by V. I. Ochkur.     

Weak Galerkin finite element methods for Parabolic equations

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H¹ and L² norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Discontinuous Galerkin methods for the linear Schrödinger equation in non-cylindrical domains

The convergence of a discontinuous Galerkin method for the linear Schrödinger equation in non-cylindrical domains of ${\mathbb{R}^m}The convergence of a discontinuous Galerkin method for the linear Schr?dinger equation in non-cylindrical domains of \mathbbR^m{\mathbb{R}^m}, m ≥ 1, is analyzed in this paper. We show the existence of the resulting approximations and prove stability and error estimates in finite element spaces of general type. When m = 1 the resulting problem is related to the standard narrow angle ‘parabolic’ approximation of the Helmholtz equation, as it appears in underwater acoustics. In this case we investigate theoretically and numerically the order of convergence using finite element spaces of piecewise polynomial functions.     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014     

Sparse finite element methods for operator equations with stochastic data

Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation Au = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments of the solution. We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment for k⩾1. The key tool in both algorithms is a “sparse tensor product” space for the approximation of with O(N(log N) ^k−1) degrees of freedom, instead of N ^k degrees of freedom for the full tensor product FEM space. A sparse Monte-Carlo FEM with M samples (i.e., deterministic solver) is proved to yield approximations to with a work of O(M N(log N) ^k−1) operations. The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples. The deterministic FEM is based on deterministic equations for in D ^k ⊂ ℝ^kd. Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of with O(N(log N) ^k−1) degrees of freedom converging at an optimal rate (up to logs). For nonlocal operators wavelet compression of the operators is used. The linear systems are solved iteratively with multilevel preconditioning. This yields an approximation for with at most O(N (log N) ^k+1) operations. This work was supported under IHP Network “Breaking Complexity” by the Swiss BBW under grant No. 02.0418     

Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h _min|)², where h _min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.     

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X_H and X_h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ? H, respectively, and a finite element space M_h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H⁵. If we choose H = O(h^2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.     

Development of a P2 element with optimal L2 convergence for biharmonic equation

It is well known that convergence rate of finite element approximation is suboptimal in the L² norm for solving biharmonic equations when P₂ or Q₂ element is used. The goal of this paper is to derive a weak Galerkin (WG) P₂ element with the L² optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle.     

A quasi-wavelet algorithm for second kind boundary integral equations

In solving integral equations with a logarithmic kernel, we combine the Galerkin approximation with periodic quasi-wavelet (PQW) [4]. We develop an algorithm for solving the integral equations with only O(N log N) arithmetic operations, where N is the number of knots. We also prove that the Galerkin approximation has a polynomial rate of convergence.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L^∞(L²) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Uniqueness results for the Minkowski problem extended to hedgehogs

The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere \mathbbSⁿ \mathbb{S}^n of ℝ ⁿ⁺¹. In this paper, we mainly consider the uniqueness question and give first results.     

Solving the Laplace-Beltrami equation on S2 using spherical triangles

The geometry of the domain −S², causes difficulty in solving the Laplace-Beltrami Equation, for example, in discretization for the differential equation. To overcome this problem, we study a numerical method, which is based on the finite element approximation with a hierarchical refinement of icosahedron for the grid. We construct a geometrically intrinsic base vector field for the Galerkin approximation. In this way, no artificial poles are introduced, and the numerical grids are distributed more evenly. We use radial projection to map the curved triangle onto a flat one, so that existing quadrature schemes can be applied for the numerical integration. The resulting system of linear algebraic equations is solved by using a conjugate gradient method. © 1996 John Wiley & Sons, Inc.     

Variational formulation for the stationary fractional advection dispersion equation

In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H^s. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations

In this article, we investigate interior penalty discontinuous Galerkin (IPDG) methods for solving a class of two‐dimensional nonlinear parabolic equations. For semi‐discrete IPDG schemes on a quasi‐uniform family of meshes, we obtain a priori bounds on solutions measured in the L² norm and in the broken Sobolev norm. The fully discrete IPDG schemes considered are based on the approximation by forward Euler difference in time and broken Sobolev space. Under a restriction related to the mesh size and time step, an hp ‐version of an a priori l^∞(L²) and l²(H¹) error estimate is derived and numerical experiments are presented.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 288–311, 2012     

Characteristics—Galerkin and mixed finite element approximation of contamination by compressible nuclear waste-disposal in porous media

A compressible nuclear waste-disposal contamination in porous media is modeled by a coupled system of partial differential equations. The approximation of this system by a Galerkin method that makes use of a modified method of characteristics for the brine, radionuclides, and heat and by a mixed finite element method for the pressure and velocity are analyzed. Optimal order error estimates are obtained. This article improves upon previous works in two aspects. First, error analysis is given with no restriction on the diffusion-dispersion tensors. That is, we have included the effects of molecular diffusion and dispersion. Secondly, optimal error estimates in H¹ and L² are derived. © 1996 John Wiley & Sons, Inc.