Generalised Galerkin methods for second order equations with significant first derivative terms

1978 ◽  
pp. 90-104 ◽  
Author(s):  
A R Mitchell ◽  
D F Griffiths
Keyword(s):  
Second Order ◽  
Galerkin Methods ◽  
First Derivative ◽  
Second Order Equations

scholarly journals Efficient spectral-Galerkin algorithms for direct solution of the integrated forms of second-order equations using ultraspherical polynomials

2007 ◽  
Vol 48 (3) ◽  
pp. 361-386 ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy
Keyword(s):  
Condition Number ◽  
Convergence Rates ◽  
Second Order ◽  
Galerkin Methods ◽  
Smooth Solutions ◽  
Ultraspherical Polynomials ◽  
Base Functions ◽  
Second Order Elliptic Equations ◽  
Second Order Equations ◽  
Dimensional Domain

AbstractIt is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) where N is the number of retained modes of polynomial approximations. This paper presents some efficient spectral algorithms, which have a condition number of O(N2), based on the ultraspherical-Galerkin methods for the integrated forms of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of Nd+1 operations for a d-dimensional domain with (N – 1)d unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

scholarly journals Boundary value problems for singular second order equations

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Alessandro Calamai ◽  
Cristina Marcelli ◽  
Francesca Papalini
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Second Order Equations
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