A multilevel approach towards unbiased sampling of random elliptic partial differential equations

Essential Spectra of Elliptic Partial Differential Equations

Let A be a closed, densely defined operator in a Banach space X. There are several definitions of the "essential" spectrum of A (cf. [ l ] , [2]). According to Wolf [3], [4] it is the complement in the complex plane of the $-set of A. The $-set $A of A is the set of points X for which (a) a(A — X), the dimension of the null space of A — X, is finite (b) R(A —X), the range of A —X, is closed (c)...

A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations

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Lecture Notes on Elliptic Partial Differential Equations

Random fractional functional differential equations

In this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the Lipschitz type condition. Moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

Multilevel First-Order System Least Squares for Nonlinear Elliptic Partial Differential Equations

A fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods. The equations are converted to a first-order system that is then linearized via Newton’s method. First-order system least squares (FOSLS) is used to formulate and discretize the Newton step, and the resulting matrix equation is solved using algebraic...