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Gradient Discretisation Method
In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless). Some core properties are required to prove the convergence of a GDM. These core properties enable complete proofs of convergence of the GDM for elliptic and parabolic problems, linear or non-linear. For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM (the quantities C_, S_ and W_, see below). For non-linear problems, the proofs are based on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the model data.J. Droniou, R. Eymard, T. Gallouët, and R. Herbin. Gradient schemes: a generic fram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mimetic Finite Difference Method And Nodal Mimetic Finite Difference Method
Mimesis (; grc, μίμησις, ''mīmēsis'') is a term used in literary criticism and philosophy that carries a wide range of meanings, including ''imitatio'', imitation, nonsensuous similarity, receptivity, representation, mimicry, the act of expression, the act of resembling, and the presentation of the self. The original Ancient Greek term ''mīmēsis'' ( grc, μίμησις, label=none) derives from ''mīmeisthai'' ( grc, μιμεῖσθαι, label=none, 'to imitate'), itself coming from ''mimos'' ( μῖμος, 'imitator, actor'). In ancient Greece, ''mīmēsis'' was an idea that governed the creation of works of art, in particular, with correspondence to the physical world understood as a model for beauty, truth, and the good. Plato contrasted ''mimesis'', or imitation, with ''diegesis'', or narrative. After Plato, the meaning of ''mimesis'' eventually shifted toward a specifically literary function in ancient Greek society. One of the best-known modern studies of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raviart–Thomas Basis Functions
In applied mathematics, Raviart–Thomas basis functions are vector basis functions used in finite element and boundary element methods. They are regularly used as basis functions when working in electromagnetics. They are sometimes called Rao-Wilton-Glisson basis functions. The space \mathrm_q spanned by the Raviart–Thomas basis functions of order q is the smallest polynomial space such that the divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ... maps \mathrm_q onto \mathrm_q, the space of piecewise polynomials of order q. Order 0 Raviart-Thomas Basis Functions in 2D In two-dimensional space, the lowest order Raviart Thomas space, \mathrm_0, has degrees of freedom on the edges of the elements of the finite element mesh. The nth edge has an associated basis function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Finite Element Method
In numerical analysis, the mixed finite element method, is a type of finite element method in which extra fields to be solved are introduced during the posing a partial differential equation problem. Somewhat related is the hybrid finite element method. The extra fields are constrained by using Lagrange multiplier fields. To be distinguished from the mixed finite element method, usual finite element methods that do not introduce such extra fields are also called irreducible or primal finite element methods. The mixed finite element method is efficient for some problems that would be numerically ill-posed if discretized by using the irreducible finite element method; one example of such problems is to compute the stress and strain fields in an almost incompressible elastic body. In mixed methods, the Lagrange multiplier fields inside the elements, usually enforcing the applicable partial differential equations. This results in a saddle point system having negative pivots and eigenval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Céa's Lemma
Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Lemma statement Let V be a real Hilbert space with the norm \, \cdot\, . Let a:V\times V\to \mathbb R be a bilinear form with the properties * , a(v, w), \le \gamma \, v\, \,\, w\, for some constant \gamma>0 and all v, w in V ( continuity) * a(v, v) \ge \alpha \, v\, ^2 for some constant \alpha>0 and all v in V ( coercivity or V-ellipticity). Let L:V\to \mathbb R be a bounded linear operator. Consider the problem of finding an element u in V such that : a(u, v)=L(v) for all v in V. Consider the same problem on a finite-dimensional subspace V_h of V, so, u_h in V_h satisfies : a(u_h, v)=L(v) for all v in V_h. By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that : \, u-u_h\, \le \frac\, u-v\, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galerkin Method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used: * Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions.A. Ern, J.L. Guermond, ''Theory and practice of finite elements'', Springer, 2004, * Bubnov–Galerkin method (after Ivan Bubnov) does not require the bilinear fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Boundary Condition
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, ''essential'' or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. The dependent unknown ''u in the same form as the weight function w'' appearing in the boundary expression is termed a ''primary variable'', and its specification constitutes the ''essential'' or Dirichlet boundary condition. The question of finding solutions to such equations is known as the Dirichlet problem. In applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition. Examples ODE For an ordinary differential equation, for instance, y'' + y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson's Equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson. Statement of the equation Poisson's equation is \Delta\varphi = f where \Delta is the Laplace operator, and f and \varphi are real or complex-valued functions on a manifold. Usually, f is given and \varphi is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as and so Poisson's equation is frequently written as \nabla^2 \varphi = f. In three-dimensional Cartesian coordinates, it takes the form \left( \frac + \frac + \frac \right)\varphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discontinuous Galerkin Method
In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics. Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a technique to numerically solve partial differential equations. In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation. The origin of the DG method for elliptic problems cannot be traced back to a single publication as features such as jump penalization in the modern sense were developed gradually. However, among the early influential contribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Mathematics
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
