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We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where $H$ is the Heaviside function and $f$ and $g$ are smooth with the assumptions that either $f$ is convex and $g$ is concave or $f$ is concave and $g$ is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with $f$ convex and $k$ of indefinite sign. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral. Some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over 1/∣J∣p-1 for 1 ≤ p≤∞ on the reference element, where J is the Jacobian of the nonaffine mapping. This allows us to introduce weak geometric conditions (depending on p) leading to interpolation error estimates in the W1,p norm, for any p ϵ [1,∞), which can be regarded as a generalization of the regular decomposition property (RDP) condition introduced in [G. Acosta and R. G. Durán, SIAM J. Numer. Anal., 38 (2000), pp. 1073-1088] for p = 2 and new RDP conditions (NRDP) for p ≠ 2. We avoid the use of the reference family elements, which allows us to extend the results to a larger class of elements and to introduce the NRDP condition in a more unified way. As far as we know, the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any p with 1 ≤ p≤∞. [ABSTRACT FROM AUTHOR]

In this paper, the frequency domain-based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation. First, the stability boundary curve is exactly determined for these methods. Then, this boundary is analyzed and compared with an accurate (ideal) boundary in different frequency ranges. Also, the critical regions in which the stability does not preserve are determined. Finally, the analytical achievements are confirmed via some numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubich's convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior. [ABSTRACT FROM AUTHOR]

In this paper, we consider the singularly perturbed reaction-diffusion problem in one and two dimensions. The boundary value problem is decomposed into a first-order system to which a suitable weighted least-squares formulation is proposed. A robust, stable, and efficient approach is developed based on local discontinuous Galerkin (LDG) discretization for the weak form. Uniform error estimates are derived. Numerical examples are presented to illustrate the method and the theoretical results. Comparison studies are made between the proposed method and other methods. [ABSTRACT FROM AUTHOR]

In this paper, we analyze convergence rates of wavelet schemes for time-dependent convection-reaction equations within the framework of the Eulerian--Lagrangian localized adjoint method (ELLAM). Under certain minimal assumptions that guarantee $ H^1 $-regularity of exact solutions, we show that a generic ELLAM scheme has a convergence rate $ \mathcal{O}(h/\sqrt{\Delta t} + \Delta t) $ in $ L^2 $-norm. Then, applying the theory of operator interpolation, we obtain error estimates for initial data with even lower regularity. Namely, it is shown that the error of such a scheme is $ \mathcal{O}((h/\sqrt{\Delta t})^\theta + (\Delta t)^\theta) $ for initial data in a Besov space $ \displaystyle B^\theta_{2,q} (0 < \theta < 1, 0 < q <= infinity) $. The error estimates are {a priori} and optimal in some cases. Numerical experiments using orthogonal wavelets are presented to illustrate the theoretical estimates. [ABSTRACT FROM AUTHOR]

The aim of this work is to solve hyperbolic conservation laws by means of a finite volume method for both spatial and time discretization. We extend the ideas developed in [X.-D. Liu and S. Osher, SIAM J. Numer. Anal., 33 (1996), pp. 760--779; X.-D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425] to fourth-order upwind and central schemes. In order to do this, once we know the cell-averages of the solution, $\overline {u}_j ^n$, in cells $I_{j}$ at time $T=t^n$, we define a new three-degree reconstruction polynomial that in each cell, $I_{j}$, presents the same shape as the cell-averages $\{ {\overline {u}_{j-1} ^n,\overline {u}_j ^n,\overline {u}_{j+1} ^n}\}$. By combining this reconstruction with the nonoscillatory property and the maximum principle requirement described in [X.-D. Liu and S. Osher, SIAM J. Numer. Anal., 33 (1996), pp. 760--779] we obtain a fourth-order scheme that satisfies the total variation bounded (TVB) property. Extension to systems is carried out by componentwise application of the scalar framework. Numerical experiments confirm the order of the schemes presented in this paper and their nonoscillatory behavior in different test problems. [ABSTRACT FROM AUTHOR]

We present in this paper numerical simulations of reactive flows interacting with shock waves. We argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long-term integrations of complicated flows, even in the presence of shock waves. A spectral code is described in detail, and the theory that leads to each of its components is explained. Results of interactions of hydrogen jets with shock waves are presented and analyzed, and comparisons with ENO finite difference schemes are carried out. [ABSTRACT FROM AUTHOR]

This paper develops a least-squares approach to the solution of the incompressible Navier­Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier­Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle based on L2 norms applied to this system yields optimal discretization error estimates in the H1 norm in each variable, including the velocity flux. An analogous principle based on the use of an H-1 norm for the reduced system (with no curl or trace constraints) is shown to yield similar estimates, but now in the L2 norm for velocity-flux and pressure. Although the H-1 least-squares principle does not allow practical implementation, these results are critical to the analysis of a practical least-squares method for the reduced system based on a discrete equivalent of the negative norm. A practical method of this type is the subject of a companion paper. Finally, we establish optimal multigrid convergence estimates for the algebraic system resulting from the L2 norm approach. [ABSTRACT FROM AUTHOR]

This paper deals with superconvergence phenomena in general grids when projectionbased approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme. [ABSTRACT FROM AUTHOR]

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

We develop a family of new interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolic-elliptic-type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original Keller-Segel model in the form of a convection-diffusionreaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten Keller-Segel system. Our methods employ the central-upwind numerical fluxes, originally developed in the context of finite-volume methods for hyperbolic systems of conservation laws. In this paper, we consider Cartesian grids and prove error estimates for the proposed high-order discontinuous Galerkin methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution. We also show that the blow-up time of the exact solution is bounded from above by the blow-up time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillation-free, even though no slope-limiting technique has been implemented. [ABSTRACT FROM AUTHOR]

We study two space-time mesh refinement methods as the one introduced in [F. Collino, T. Fouquet, and P. Joly, Numer. Math., 95 (2003), pp. 197--221]. The stability of such methods is guaranteed by construction through the conservation of a discrete energy. In this paper, we show the L2 convergence of these schemes and provide optimal error estimates. The proof is based on energy techniques and bootstrap arguments. Our results are validated with numerical simulations and compared with results from plane wave analysis [F. Collino, T. Fouquet, and P. Joly, Numer. Math., 95 (2003), pp. 223--251]. [ABSTRACT FROM AUTHOR]

A finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math., 79 (1998), pp. 175--202]. In this paper, we have used a finite element discretization, where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. While the continuous time Galerkin method is discussed for the spatially discrete scheme, the discontinuous Galerkin method is analyzed for the fully discrete scheme. The interfaces and boundaries of the domains are assumed to be smooth for our purpose. [ABSTRACT FROM AUTHOR]

A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green's functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multi-dimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented. [ABSTRACT FROM AUTHOR]

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for cis with a linear advection equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem. [ABSTRACT FROM AUTHOR]

In this paper we describe techniques to represent data which originate from discretization of functions in unstructured meshes in terms of their local scale components. To do so we consider a nested sequence of discretization, which corresponds to increasing levels of resolution, and we define the scales as the ‘difference in information’ between any two successive levels. We obtain data compression by eliminating scale-coefficients which are sufficiently small. This capability for data compression can be used to reduce the cost of numerical schemes by solving for the more compact representation of the numerical solution in terms of its significant scale-coefficients. [ABSTRACT FROM AUTHOR]

A numerical radius approach is taken in this paper in order to discuss sufficient stability conditions for a well-known family of finite difference schemes for the initial value problem associated with the Petrowski well-posed, multispace-dimensional parabolic system [This symbol cannot be presented in ASCII format] where Apq, Bp, and C are constant matrices, Apq being Hermitian. [ABSTRACT FROM AUTHOR]

A cell-centered finite volume method is proposed to approximate numerically the solution to the steady convection-diffusion equation on unstructured meshes of $d$-simplexes, where $d\geq 2$ is the spatial dimension. The method is formally second-order accurate by means of a piecewise linear reconstruction within each cell and at mesh vertices. An algorithm is provided to calculate nonnegative and bounded weights. Face gradients, required to discretize the diffusive fluxes, are defined by a nonlinear strategy that allows us to demonstrate the existence of a maximum principle. Finally, a set of numerical results documents the performance of the method in treating problems with internal layers and solutions with strong gradients. [ABSTRACT FROM AUTHOR]

We develop an adaptive perfectly matched layer (PML) technique for solving the time-harmonic scattering problems. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. The derived finite element a posteriori estimate for adapting meshes has the nice feature that it decays exponentially away from the boundary of the fixed domain where the PML layer is placed. This property makes the total computational costs insensitive to the thickness of the PML absorbing layers. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method. [ABSTRACT FROM AUTHOR]

We obtain error bounds for monotone approximation schemes of Hamilton--Jacobi--Bellman equations. These bounds improve previous results of Krylov and the authors. The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton--Jacobi--Bellman equation. [ABSTRACT FROM AUTHOR]

A new mixed finite element method is formulated for the Stokes equations, in which the two components of the velocity and the pressure are defined on different meshes. First-order error estimates are obtained for both the velocity and the pressure. Also, the well-known MAC method is derived from the resulting finite element method. [ABSTRACT FROM AUTHOR]

A recent result in [B. T. Helenbrook, SIAM J. Numer. Anal., 47 (2009), pp. 1304-1318] on the nonexistence of Gauss-Lobatto cubature rules on the triangle is strengthened by establishing a lower bound for the number of nodes of such rules. A method of constructing Lobatto-type cubature rules on the triangle is given and used to construct several examples. [ABSTRACT FROM AUTHOR]

A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional local discontinuous Galerkin method, the scheme called the direct discontinuous Galerkin (DDG) method is based on the direct weak formulation for solutions of parabolic equations in each computational cell and lets cells communicate via the numerical flux ux only. We propose a general numerical flux formula for the solution derivative, which is consistent and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the 1D linear case, kth order accuracy in an energy norm is proven when using kth degree polynomials. The DDG method has the advantage of easier formulation and implementation and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes. [ABSTRACT FROM AUTHOR]

We present new local anisotropic error estimates for the Lagrangian finite element interpolation. The bounds apply to affine equivalent elements and use information from directional derivatives of the function to interpolate along a set of adjacent edges. These new bounds do not require any geometric limitation but may vary, in some cases, with the node ordering. Several existing results are recovered from the new bounds. Examples compare the asymptotic behavior of the new and existing bounds when the diameter of the element goes to zero. For some elements with small or large angles, our new bound exhibits the same asymptotic behavior as the norm of the interpolation error while existing results do not have the correct asymptotic behavior. [ABSTRACT FROM AUTHOR]

Thermomagnetic recording uses local heating of ferromagnetic media to locally decrease coercivity and change saturation magnetization of the material and alleviate magnetization reversal. A modified Landau-Lifshitz equation is proposed as a phenomenological model to allow for changes in magnetization magnitude and is studied both analytically and numerically. [ABSTRACT FROM AUTHOR]

We consider DG-methods for second order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric versions of the DG-method have regular system matrices without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a DG-method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and nonsymmetric DG-method without stabilization. All of these proposed methods share the feature that they conserve mass locally independent of the penalty parameter. [ABSTRACT FROM AUTHOR]

We study the convergence of an adaptive interior penalty discontinuous Galerkin (IPDG) method for a two-dimensional model second order elliptic boundary value problem. Based on a residual-type a posteriori error estimator, we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the meshdependent energy norm associated with the IPDG method. In contrast to recent work on adaptive IPDG methods [O. Karakashian and F. Pascal, Convergence of Adaptive Discontinuous Galerkin Approximations of Second-order Elliptic Problems, preprint, University of Tennessee, Knoxville, TN, 2007], the convergence analysis does not require multiple interior nodes for refined elements of the triangulation. In fact, it will be shown that bisection of the elements is sufficient. The main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to the energy inner product. The results of numerical experiments are given to illustrate the performance of the adaptive method. [ABSTRACT FROM AUTHOR]

We consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient Stokes equation. The analysis is valid for a class of symmetric pressure stabilization operators, but also for standard, inf-sup stable, velocity/pressure spaces without stabilization. Provided the initial data are chosen as a specific (method-dependent) Ritz-projection, we get unconditional stability and optimal convergence for both pressure and velocity approximations, in natural norms. For arbitrary interpolations of the initial data, a condition between the space and time discretization parameters has to be verified in order to guarantee pressure stability. [ABSTRACT FROM AUTHOR]

A mixed spectral method is proposed and analyzed for the Stokes problem in a semi-infinite channel. The method is based on a generalized Galerkin approximation with Laguerre functions in the x direction and Legendre polynomials in the y direction. The well-posedness of this method is established by deriving a lower bound on the inf-sup constant. Numerical results indicate that the derived lower bound is sharp. Rigorous error analysis is also carried out. [ABSTRACT FROM AUTHOR]

We develop the finite element method for the numerical resolution of the space and time fractional Fokker-Planck equation, which is an effective tool for describing a process with both traps and flights; the time fractional derivative of the equation is used to characterize the traps, and the flights are depicted by the space fractional derivative. The stability and error estimates are rigorously established, and we prove that the convergent order is O(k2-α +hμ), where k is the time step size and h the space step size. Numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims. [ABSTRACT FROM AUTHOR]

A BDDC (balancing domain decomposition by constraints) method is developed for elliptic equations, with discontinuous coefficients, discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face, which is the intersection of the boundaries of a pair of subdomains. A condition number bound of the form C maxi{(1 + log(Hi/hi))2} is established under certain assumptions on the geometrically nonconforming subdomain partition in the three-dimensional case. Here Hi and hi are the subdomain diameters and the mesh sizes, respectively. In the geometrically conforming case and the geometrically nonconforming cases in two dimensions, no assumptions on the subdomain partition are required. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet version of the FETI-DP algorithm. The results are illustrated by numerical experiments which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Compressible flows in a nozzle can be modeled by the gas dynamics equations in one-dimensional space with source terms. It turns out that along stationary waves, the entropy is conserved. Investigating properties of the system leads us to the determination of stationary waves. Relying on this analysis, we construct a numerical scheme which takes into account the use of stationary waves. Our scheme is shown to be capable of maintaining equilibrium states. This demonstrates the efficiency of the new scheme over classical ones, which usually give unsatisfactory results when reducing the refinement of the mesh-size. Moreover, our scheme converges much faster than the classical ones in most cases. [ABSTRACT FROM AUTHOR]

The binary space partition (BSP) technique is a simple and efficient method to adaptively partition an initial given domain to match the geometry of a given input function. As such, the BSP technique has been widely used by practitioners, but up until now no rigorous mathematical justification for it has been offered. Here we attempt to put the technique on sound mathematical foundations, and we offer an enhancement of the BSP algorithm in the spirit of what we are going to call geometric wavelets. This new approach to sparse geometric representation is based on recent developments in the theory of multivariate nonlinear piecewise polynomial approximation. We provide numerical examples of n-term geometric wavelet approximations of known test images and compare them with dyadic wavelet approximation. We also discuss applications to image denoising and compression. [ABSTRACT FROM AUTHOR]

Numerical computations show that solutions of hyperbolic conservation laws obtained by second or higher order shock capturing methods in many cases are only first order accurate downstream of shocks (see, e.g., [M. H. Carpenter and J. H. Casper, AIAA J., 37 (1999), pp. 1072--1079]). We use matched asymptotic expansions to analyze the degeneration in order of accuracy for stationary solutions of hyperbolic conservation laws in two space dimensions. [ABSTRACT FROM AUTHOR]

We prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in H1- and L2-norms. We present two proofs: one uses the standard L2-projection and the other relies on a new, weighted Clément-type interpolation operator. [ABSTRACT FROM AUTHOR]

A least-squares finite element method for second-order elliptic boundary value problems having interfaces due to discontinuous media properties is proposed and analyzed. Both Dirichlet and Neumann boundary data are treated. The boundary value problems are recast into a first-order formulation to which a suitable least-squares principle is applied. Among the advantages of the method are that nonconforming, with respect to the interface, approximating subspaces may be used. Moreover, the grids used on each side of an interface need not coincide along the interface. Error estimates are derived that improve on other treatments of interface problems and a numerical example is provided to illustrate the method and the analyses. [ABSTRACT FROM AUTHOR]