DiscretizationIn applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).
Finite element methodThe 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).
Finite difference methodIn numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
Finite differenceA finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.
Numerical stabilityIn the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Earth's crustEarth's crust is Earth's thick outer shell of rock, referring to less than 1% of the Earth's radius and volume. It is the top component of the lithosphere, a division of Earth's layers that includes the crust and the upper part of the mantle. The lithosphere is broken into tectonic plates whose motion allows heat to escape from the interior of the Earth into space. The crust lies on top of the mantle, a configuration that is stable because the upper mantle is made of peridotite and is therefore significantly denser than the crust.
Discrete mathematicsDiscrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry.
Continental crustContinental crust is the layer of igneous, metamorphic, and sedimentary rocks that forms the geological continents and the areas of shallow seabed close to their shores, known as continental shelves. This layer is sometimes called sial because its bulk composition is richer in aluminium silicates (Al-Si) and has a lower density compared to the oceanic crust, called sima which is richer in magnesium silicate (Mg-Si) minerals.
Oceanic crustOceanic crust is the uppermost layer of the oceanic portion of the tectonic plates. It is composed of the upper oceanic crust, with pillow lavas and a dike complex, and the lower oceanic crust, composed of troctolite, gabbro and ultramafic cumulates. The crust overlies the rigid uppermost layer of the mantle. The crust and the rigid upper mantle layer together constitute oceanic lithosphere. Oceanic crust is primarily composed of mafic rocks, or sima, which is rich in iron and magnesium.
Numerical analysisNumerical 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.
